{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0001","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2],[1,2,0],[2,0,1]],\"source_identity\":0,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"target_identity\":0,\"mapping\":[0,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [0, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 2, 2, 是], [0, 2, 1, 1, 是], [1, 0, 2, 2, 是], [1, 1, 1, 1, 是], [1, 2, 0, 0, 是], [2, 0, 1, 1, 是], [2, 1, 0, 0, 是], [2, 2, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(3, 3, [0, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 2, 2, 是], [0, 2, 1, 1, 是], [1, 0, 2, 2, 是], [1, 1, 1, 1, 是], [1, 2, 0, 0, 是], [2, 0, 1, 1, 是], [2, 1, 0, 0, 是], [2, 2, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141001,"parameters":{"source_table":[[0,1,2],[1,2,0],[2,0,1]],"source_identity":0,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"target_identity":0,"mapping":[0,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,2,1],\"source_identity\":0,\"source_table\":[[0,1,2],[1,2,0],[2,0,1]],\"target_identity\":0,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0002","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"source_identity\":0,\"target_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"mapping\":[0,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [0, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 3, 3, 是], [0, 2, 2, 2, 是], [0, 3, 1, 1, 是], [1, 0, 3, 3, 是], [1, 1, 2, 2, 是], [1, 2, 1, 1, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是], [2, 3, 3, 3, 是], [3, 0, 1, 1, 是], [3, 1, 0, 0, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(4, 4, [0, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 3, 3, 是], [0, 2, 2, 2, 是], [0, 3, 1, 1, 是], [1, 0, 3, 3, 是], [1, 1, 2, 2, 是], [1, 2, 1, 1, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是], [2, 3, 3, 3, 是], [3, 0, 1, 1, 是], [3, 1, 0, 0, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141002,"parameters":{"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"target_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0003","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_identity\":0,\"mapping\":[0,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [0, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 4, 4, 是], [0, 2, 3, 3, 是], [0, 3, 2, 2, 是], [0, 4, 1, 1, 是], [1, 0, 4, 4, 是], [1, 1, 3, 3, 是], [1, 2, 2, 2, 是], [1, 3, 1, 1, 是], [1, 4, 0, 0, 是], [2, 0, 3, 3, 是], [2, 1, 2, 2, 是], [2, 2, 1, 1, 是], [2, 3, 0, 0, 是], [2, 4, 4, 4, 是], [3, 0, 2, 2, 是], [3, 1, 1, 1, 是], [3, 2, 0, 0, 是], [3, 3, 4, 4, 是], [3, 4, 3, 3, 是], [4, 0, 1, 1, 是], [4, 1, 0, 0, 是], [4, 2, 4, 4, 是], [4, 3, 3, 3, 是], [4, 4, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(5, 5, [0, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 4, 4, 是], [0, 2, 3, 3, 是], [0, 3, 2, 2, 是], [0, 4, 1, 1, 是], [1, 0, 4, 4, 是], [1, 1, 3, 3, 是], [1, 2, 2, 2, 是], [1, 3, 1, 1, 是], [1, 4, 0, 0, 是], [2, 0, 3, 3, 是], [2, 1, 2, 2, 是], [2, 2, 1, 1, 是], [2, 3, 0, 0, 是], [2, 4, 4, 4, 是], [3, 0, 2, 2, 是], [3, 1, 1, 1, 是], [3, 2, 0, 0, 是], [3, 3, 4, 4, 是], [3, 4, 3, 3, 是], [4, 0, 1, 1, 是], [4, 1, 0, 0, 是], [4, 2, 4, 4, 是], [4, 3, 3, 3, 是], [4, 4, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141003,"parameters":{"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0004","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"mapping\":[0,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 6, [0, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 5, 5, 是], [0, 2, 4, 4, 是], [0, 3, 3, 3, 是], [0, 4, 2, 2, 是], [0, 5, 1, 1, 是], [1, 0, 5, 5, 是], [1, 1, 4, 4, 是], [1, 2, 3, 3, 是], [1, 3, 2, 2, 是], [1, 4, 1, 1, 是], [1, 5, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 3, 3, 是], [2, 2, 2, 2, 是], [2, 3, 1, 1, 是], [2, 4, 0, 0, 是], [2, 5, 5, 5, 是], [3, 0, 3, 3, 是], [3, 1, 2, 2, 是], [3, 2, 1, 1, 是], [3, 3, 0, 0, 是], [3, 4, 5, 5, 是], [3, 5, 4, 4, 是], [4, 0, 2, 2, 是], [4, 1, 1, 1, 是], [4, 2, 0, 0, 是], [4, 3, 5, 5, 是], [4, 4, 4, 4, 是], [4, 5, 3, 3, 是], [5, 0, 1, 1, 是], [5, 1, 0, 0, 是], [5, 2, 5, 5, 是], [5, 3, 4, 4, 是], [5, 4, 3, 3, 是], [5, 5, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(6, 6, [0, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 5, 5, 是], [0, 2, 4, 4, 是], [0, 3, 3, 3, 是], [0, 4, 2, 2, 是], [0, 5, 1, 1, 是], [1, 0, 5, 5, 是], [1, 1, 4, 4, 是], [1, 2, 3, 3, 是], [1, 3, 2, 2, 是], [1, 4, 1, 1, 是], [1, 5, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 3, 3, 是], [2, 2, 2, 2, 是], [2, 3, 1, 1, 是], [2, 4, 0, 0, 是], [2, 5, 5, 5, 是], [3, 0, 3, 3, 是], [3, 1, 2, 2, 是], [3, 2, 1, 1, 是], [3, 3, 0, 0, 是], [3, 4, 5, 5, 是], [3, 5, 4, 4, 是], [4, 0, 2, 2, 是], [4, 1, 1, 1, 是], [4, 2, 0, 0, 是], [4, 3, 5, 5, 是], [4, 4, 4, 4, 是], [4, 5, 3, 3, 是], [5, 0, 1, 1, 是], [5, 1, 0, 0, 是], [5, 2, 5, 5, 是], [5, 3, 4, 4, 是], [5, 4, 3, 3, 是], [5, 5, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141004,"parameters":{"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0005","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],\"target_identity\":0,\"mapping\":[0,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 7, [0, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 6, 6, 是], [0, 2, 5, 5, 是], [0, 3, 4, 4, 是], [0, 4, 3, 3, 是], [0, 5, 2, 2, 是], [0, 6, 1, 1, 是], [1, 0, 6, 6, 是], [1, 1, 5, 5, 是], [1, 2, 4, 4, 是], [1, 3, 3, 3, 是], [1, 4, 2, 2, 是], [1, 5, 1, 1, 是], [1, 6, 0, 0, 是], [2, 0, 5, 5, 是], [2, 1, 4, 4, 是], [2, 2, 3, 3, 是], [2, 3, 2, 2, 是], [2, 4, 1, 1, 是], [2, 5, 0, 0, 是], [2, 6, 6, 6, 是], [3, 0, 4, 4, 是], [3, 1, 3, 3, 是], [3, 2, 2, 2, 是], [3, 3, 1, 1, 是], [3, 4, 0, 0, 是], [3, 5, 6, 6, 是], [3, 6, 5, 5, 是], [4, 0, 3, 3, 是], [4, 1, 2, 2, 是], [4, 2, 1, 1, 是], [4, 3, 0, 0, 是], [4, 4, 6, 6, 是], [4, 5, 5, 5, 是], [4, 6, 4, 4, 是], [5, 0, 2, 2, 是], [5, 1, 1, 1, 是], [5, 2, 0, 0, 是], [5, 3, 6, 6, 是], [5, 4, 5, 5, 是], [5, 5, 4, 4, 是], [5, 6, 3, 3, 是], [6, 0, 1, 1, 是], [6, 1, 0, 0, 是], [6, 2, 6, 6, 是], [6, 3, 5, 5, 是], [6, 4, 4, 4, 是], [6, 5, 3, 3, 是], [6, 6, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(7, 7, [0, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 6, 6, 是], [0, 2, 5, 5, 是], [0, 3, 4, 4, 是], [0, 4, 3, 3, 是], [0, 5, 2, 2, 是], [0, 6, 1, 1, 是], [1, 0, 6, 6, 是], [1, 1, 5, 5, 是], [1, 2, 4, 4, 是], [1, 3, 3, 3, 是], [1, 4, 2, 2, 是], [1, 5, 1, 1, 是], [1, 6, 0, 0, 是], [2, 0, 5, 5, 是], [2, 1, 4, 4, 是], [2, 2, 3, 3, 是], [2, 3, 2, 2, 是], [2, 4, 1, 1, 是], [2, 5, 0, 0, 是], [2, 6, 6, 6, 是], [3, 0, 4, 4, 是], [3, 1, 3, 3, 是], [3, 2, 2, 2, 是], [3, 3, 1, 1, 是], [3, 4, 0, 0, 是], [3, 5, 6, 6, 是], [3, 6, 5, 5, 是], [4, 0, 3, 3, 是], [4, 1, 2, 2, 是], [4, 2, 1, 1, 是], [4, 3, 0, 0, 是], [4, 4, 6, 6, 是], [4, 5, 5, 5, 是], [4, 6, 4, 4, 是], [5, 0, 2, 2, 是], [5, 1, 1, 1, 是], [5, 2, 0, 0, 是], [5, 3, 6, 6, 是], [5, 4, 5, 5, 是], [5, 5, 4, 4, 是], [5, 6, 3, 3, 是], [6, 0, 1, 1, 是], [6, 1, 0, 0, 是], [6, 2, 6, 6, 是], [6, 3, 5, 5, 是], [6, 4, 4, 4, 是], [6, 5, 3, 3, 是], [6, 6, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141005,"parameters":{"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0006","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"mapping\":[0,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 8, [0, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 7, 7, 是], [0, 2, 6, 6, 是], [0, 3, 5, 5, 是], [0, 4, 4, 4, 是], [0, 5, 3, 3, 是], [0, 6, 2, 2, 是], [0, 7, 1, 1, 是], [1, 0, 7, 7, 是], [1, 1, 6, 6, 是], [1, 2, 5, 5, 是], [1, 3, 4, 4, 是], [1, 4, 3, 3, 是], [1, 5, 2, 2, 是], [1, 6, 1, 1, 是], [1, 7, 0, 0, 是], [2, 0, 6, 6, 是], [2, 1, 5, 5, 是], [2, 2, 4, 4, 是], [2, 3, 3, 3, 是], [2, 4, 2, 2, 是], [2, 5, 1, 1, 是], [2, 6, 0, 0, 是], [2, 7, 7, 7, 是], [3, 0, 5, 5, 是], [3, 1, 4, 4, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是], [3, 4, 1, 1, 是], [3, 5, 0, 0, 是], [3, 6, 7, 7, 是], [3, 7, 6, 6, 是], [4, 0, 4, 4, 是], [4, 1, 3, 3, 是], [4, 2, 2, 2, 是], [4, 3, 1, 1, 是], [4, 4, 0, 0, 是], [4, 5, 7, 7, 是], [4, 6, 6, 6, 是], [4, 7, 5, 5, 是], [5, 0, 3, 3, 是], [5, 1, 2, 2, 是], [5, 2, 1, 1, 是], [5, 3, 0, 0, 是], [5, 4, 7, 7, 是], [5, 5, 6, 6, 是], [5, 6, 5, 5, 是], [5, 7, 4, 4, 是], [6, 0, 2, 2, 是], [6, 1, 1, 1, 是], [6, 2, 0, 0, 是], [6, 3, 7, 7, 是], [6, 4, 6, 6, 是], [6, 5, 5, 5, 是], [6, 6, 4, 4, 是], [6, 7, 3, 3, 是], [7, 0, 1, 1, 是], [7, 1, 0, 0, 是], [7, 2, 7, 7, 是], [7, 3, 6, 6, 是], [7, 4, 5, 5, 是], [7, 5, 4, 4, 是], [7, 6, 3, 3, 是], [7, 7, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(8, 8, [0, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 7, 7, 是], [0, 2, 6, 6, 是], [0, 3, 5, 5, 是], [0, 4, 4, 4, 是], [0, 5, 3, 3, 是], [0, 6, 2, 2, 是], [0, 7, 1, 1, 是], [1, 0, 7, 7, 是], [1, 1, 6, 6, 是], [1, 2, 5, 5, 是], [1, 3, 4, 4, 是], [1, 4, 3, 3, 是], [1, 5, 2, 2, 是], [1, 6, 1, 1, 是], [1, 7, 0, 0, 是], [2, 0, 6, 6, 是], [2, 1, 5, 5, 是], [2, 2, 4, 4, 是], [2, 3, 3, 3, 是], [2, 4, 2, 2, 是], [2, 5, 1, 1, 是], [2, 6, 0, 0, 是], [2, 7, 7, 7, 是], [3, 0, 5, 5, 是], [3, 1, 4, 4, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是], [3, 4, 1, 1, 是], [3, 5, 0, 0, 是], [3, 6, 7, 7, 是], [3, 7, 6, 6, 是], [4, 0, 4, 4, 是], [4, 1, 3, 3, 是], [4, 2, 2, 2, 是], [4, 3, 1, 1, 是], [4, 4, 0, 0, 是], [4, 5, 7, 7, 是], [4, 6, 6, 6, 是], [4, 7, 5, 5, 是], [5, 0, 3, 3, 是], [5, 1, 2, 2, 是], [5, 2, 1, 1, 是], [5, 3, 0, 0, 是], [5, 4, 7, 7, 是], [5, 5, 6, 6, 是], [5, 6, 5, 5, 是], [5, 7, 4, 4, 是], [6, 0, 2, 2, 是], [6, 1, 1, 1, 是], [6, 2, 0, 0, 是], [6, 3, 7, 7, 是], [6, 4, 6, 6, 是], [6, 5, 5, 5, 是], [6, 6, 4, 4, 是], [6, 7, 3, 3, 是], [7, 0, 1, 1, 是], [7, 1, 0, 0, 是], [7, 2, 7, 7, 是], [7, 3, 6, 6, 是], [7, 4, 5, 5, 是], [7, 5, 4, 4, 是], [7, 6, 3, 3, 是], [7, 7, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141006,"parameters":{"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0007","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],\"target_identity\":0,\"mapping\":[0,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 9, [0, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 8, 8, 是], [0, 2, 7, 7, 是], [0, 3, 6, 6, 是], [0, 4, 5, 5, 是], [0, 5, 4, 4, 是], [0, 6, 3, 3, 是], [0, 7, 2, 2, 是], [0, 8, 1, 1, 是], [1, 0, 8, 8, 是], [1, 1, 7, 7, 是], [1, 2, 6, 6, 是], [1, 3, 5, 5, 是], [1, 4, 4, 4, 是], [1, 5, 3, 3, 是], [1, 6, 2, 2, 是], [1, 7, 1, 1, 是], [1, 8, 0, 0, 是], [2, 0, 7, 7, 是], [2, 1, 6, 6, 是], [2, 2, 5, 5, 是], [2, 3, 4, 4, 是], [2, 4, 3, 3, 是], [2, 5, 2, 2, 是], [2, 6, 1, 1, 是], [2, 7, 0, 0, 是], [2, 8, 8, 8, 是], [3, 0, 6, 6, 是], [3, 1, 5, 5, 是], [3, 2, 4, 4, 是], [3, 3, 3, 3, 是], [3, 4, 2, 2, 是], [3, 5, 1, 1, 是], [3, 6, 0, 0, 是], [3, 7, 8, 8, 是], [3, 8, 7, 7, 是], [4, 0, 5, 5, 是], [4, 1, 4, 4, 是], [4, 2, 3, 3, 是], [4, 3, 2, 2, 是], [4, 4, 1, 1, 是], [4, 5, 0, 0, 是], [4, 6, 8, 8, 是], [4, 7, 7, 7, 是], [4, 8, 6, 6, 是], [5, 0, 4, 4, 是], [5, 1, 3, 3, 是], [5, 2, 2, 2, 是], [5, 3, 1, 1, 是], [5, 4, 0, 0, 是], [5, 5, 8, 8, 是], [5, 6, 7, 7, 是], [5, 7, 6, 6, 是], [5, 8, 5, 5, 是], [6, 0, 3, 3, 是], [6, 1, 2, 2, 是], [6, 2, 1, 1, 是], [6, 3, 0, 0, 是], [6, 4, 8, 8, 是], [6, 5, 7, 7, 是], [6, 6, 6, 6, 是], [6, 7, 5, 5, 是], [6, 8, 4, 4, 是], [7, 0, 2, 2, 是], [7, 1, 1, 1, 是], [7, 2, 0, 0, 是], [7, 3, 8, 8, 是], [7, 4, 7, 7, 是], [7, 5, 6, 6, 是], [7, 6, 5, 5, 是], [7, 7, 4, 4, 是], [7, 8, 3, 3, 是], [8, 0, 1, 1, 是], [8, 1, 0, 0, 是], [8, 2, 8, 8, 是], [8, 3, 7, 7, 是], [8, 4, 6, 6, 是], [8, 5, 5, 5, 是], [8, 6, 4, 4, 是], [8, 7, 3, 3, 是], [8, 8, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(9, 9, [0, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 8, 8, 是], [0, 2, 7, 7, 是], [0, 3, 6, 6, 是], [0, 4, 5, 5, 是], [0, 5, 4, 4, 是], [0, 6, 3, 3, 是], [0, 7, 2, 2, 是], [0, 8, 1, 1, 是], [1, 0, 8, 8, 是], [1, 1, 7, 7, 是], [1, 2, 6, 6, 是], [1, 3, 5, 5, 是], [1, 4, 4, 4, 是], [1, 5, 3, 3, 是], [1, 6, 2, 2, 是], [1, 7, 1, 1, 是], [1, 8, 0, 0, 是], [2, 0, 7, 7, 是], [2, 1, 6, 6, 是], [2, 2, 5, 5, 是], [2, 3, 4, 4, 是], [2, 4, 3, 3, 是], [2, 5, 2, 2, 是], [2, 6, 1, 1, 是], [2, 7, 0, 0, 是], [2, 8, 8, 8, 是], [3, 0, 6, 6, 是], [3, 1, 5, 5, 是], [3, 2, 4, 4, 是], [3, 3, 3, 3, 是], [3, 4, 2, 2, 是], [3, 5, 1, 1, 是], [3, 6, 0, 0, 是], [3, 7, 8, 8, 是], [3, 8, 7, 7, 是], [4, 0, 5, 5, 是], [4, 1, 4, 4, 是], [4, 2, 3, 3, 是], [4, 3, 2, 2, 是], [4, 4, 1, 1, 是], [4, 5, 0, 0, 是], [4, 6, 8, 8, 是], [4, 7, 7, 7, 是], [4, 8, 6, 6, 是], [5, 0, 4, 4, 是], [5, 1, 3, 3, 是], [5, 2, 2, 2, 是], [5, 3, 1, 1, 是], [5, 4, 0, 0, 是], [5, 5, 8, 8, 是], [5, 6, 7, 7, 是], [5, 7, 6, 6, 是], [5, 8, 5, 5, 是], [6, 0, 3, 3, 是], [6, 1, 2, 2, 是], [6, 2, 1, 1, 是], [6, 3, 0, 0, 是], [6, 4, 8, 8, 是], [6, 5, 7, 7, 是], [6, 6, 6, 6, 是], [6, 7, 5, 5, 是], [6, 8, 4, 4, 是], [7, 0, 2, 2, 是], [7, 1, 1, 1, 是], [7, 2, 0, 0, 是], [7, 3, 8, 8, 是], [7, 4, 7, 7, 是], [7, 5, 6, 6, 是], [7, 6, 5, 5, 是], [7, 7, 4, 4, 是], [7, 8, 3, 3, 是], [8, 0, 1, 1, 是], [8, 1, 0, 0, 是], [8, 2, 8, 8, 是], [8, 3, 7, 7, 是], [8, 4, 6, 6, 是], [8, 5, 5, 5, 是], [8, 6, 4, 4, 是], [8, 7, 3, 3, 是], [8, 8, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141007,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],"target_identity":0,"mapping":[0,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0008","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"mapping\":[0,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 10, [0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 9, 9, 是], [0, 2, 8, 8, 是], [0, 3, 7, 7, 是], [0, 4, 6, 6, 是], [0, 5, 5, 5, 是], [0, 6, 4, 4, 是], [0, 7, 3, 3, 是], [0, 8, 2, 2, 是], [0, 9, 1, 1, 是], [1, 0, 9, 9, 是], [1, 1, 8, 8, 是], [1, 2, 7, 7, 是], [1, 3, 6, 6, 是], [1, 4, 5, 5, 是], [1, 5, 4, 4, 是], [1, 6, 3, 3, 是], [1, 7, 2, 2, 是], [1, 8, 1, 1, 是], [1, 9, 0, 0, 是], [2, 0, 8, 8, 是], [2, 1, 7, 7, 是], [2, 2, 6, 6, 是], [2, 3, 5, 5, 是], [2, 4, 4, 4, 是], [2, 5, 3, 3, 是], [2, 6, 2, 2, 是], [2, 7, 1, 1, 是], [2, 8, 0, 0, 是], [2, 9, 9, 9, 是], [3, 0, 7, 7, 是], [3, 1, 6, 6, 是], [3, 2, 5, 5, 是], [3, 3, 4, 4, 是], [3, 4, 3, 3, 是], [3, 5, 2, 2, 是], [3, 6, 1, 1, 是], [3, 7, 0, 0, 是], [3, 8, 9, 9, 是], [3, 9, 8, 8, 是], [4, 0, 6, 6, 是], [4, 1, 5, 5, 是], [4, 2, 4, 4, 是], [4, 3, 3, 3, 是], [4, 4, 2, 2, 是], [4, 5, 1, 1, 是], [4, 6, 0, 0, 是], [4, 7, 9, 9, 是], [4, 8, 8, 8, 是], [4, 9, 7, 7, 是], [5, 0, 5, 5, 是], [5, 1, 4, 4, 是], [5, 2, 3, 3, 是], [5, 3, 2, 2, 是], [5, 4, 1, 1, 是], [5, 5, 0, 0, 是], [5, 6, 9, 9, 是], [5, 7, 8, 8, 是], [5, 8, 7, 7, 是], [5, 9, 6, 6, 是], [6, 0, 4, 4, 是], [6, 1, 3, 3, 是], [6, 2, 2, 2, 是], [6, 3, 1, 1, 是], [6, 4, 0, 0, 是], [6, 5, 9, 9, 是], [6, 6, 8, 8, 是], [6, 7, 7, 7, 是], [6, 8, 6, 6, 是], [6, 9, 5, 5, 是], [7, 0, 3, 3, 是], [7, 1, 2, 2, 是], [7, 2, 1, 1, 是], [7, 3, 0, 0, 是], [7, 4, 9, 9, 是], [7, 5, 8, 8, 是], [7, 6, 7, 7, 是], [7, 7, 6, 6, 是], [7, 8, 5, 5, 是], [7, 9, 4, 4, 是], [8, 0, 2, 2, 是], [8, 1, 1, 1, 是], [8, 2, 0, 0, 是], [8, 3, 9, 9, 是], [8, 4, 8, 8, 是], [8, 5, 7, 7, 是], [8, 6, 6, 6, 是], [8, 7, 5, 5, 是], [8, 8, 4, 4, 是], [8, 9, 3, 3, 是], [9, 0, 1, 1, 是], [9, 1, 0, 0, 是], [9, 2, 9, 9, 是], [9, 3, 8, 8, 是], [9, 4, 7, 7, 是], [9, 5, 6, 6, 是], [9, 6, 5, 5, 是], [9, 7, 4, 4, 是], [9, 8, 3, 3, 是], [9, 9, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(10, 10, [0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 9, 9, 是], [0, 2, 8, 8, 是], [0, 3, 7, 7, 是], [0, 4, 6, 6, 是], [0, 5, 5, 5, 是], [0, 6, 4, 4, 是], [0, 7, 3, 3, 是], [0, 8, 2, 2, 是], [0, 9, 1, 1, 是], [1, 0, 9, 9, 是], [1, 1, 8, 8, 是], [1, 2, 7, 7, 是], [1, 3, 6, 6, 是], [1, 4, 5, 5, 是], [1, 5, 4, 4, 是], [1, 6, 3, 3, 是], [1, 7, 2, 2, 是], [1, 8, 1, 1, 是], [1, 9, 0, 0, 是], [2, 0, 8, 8, 是], [2, 1, 7, 7, 是], [2, 2, 6, 6, 是], [2, 3, 5, 5, 是], [2, 4, 4, 4, 是], [2, 5, 3, 3, 是], [2, 6, 2, 2, 是], [2, 7, 1, 1, 是], [2, 8, 0, 0, 是], [2, 9, 9, 9, 是], [3, 0, 7, 7, 是], [3, 1, 6, 6, 是], [3, 2, 5, 5, 是], [3, 3, 4, 4, 是], [3, 4, 3, 3, 是], [3, 5, 2, 2, 是], [3, 6, 1, 1, 是], [3, 7, 0, 0, 是], [3, 8, 9, 9, 是], [3, 9, 8, 8, 是], [4, 0, 6, 6, 是], [4, 1, 5, 5, 是], [4, 2, 4, 4, 是], [4, 3, 3, 3, 是], [4, 4, 2, 2, 是], [4, 5, 1, 1, 是], [4, 6, 0, 0, 是], [4, 7, 9, 9, 是], [4, 8, 8, 8, 是], [4, 9, 7, 7, 是], [5, 0, 5, 5, 是], [5, 1, 4, 4, 是], [5, 2, 3, 3, 是], [5, 3, 2, 2, 是], [5, 4, 1, 1, 是], [5, 5, 0, 0, 是], [5, 6, 9, 9, 是], [5, 7, 8, 8, 是], [5, 8, 7, 7, 是], [5, 9, 6, 6, 是], [6, 0, 4, 4, 是], [6, 1, 3, 3, 是], [6, 2, 2, 2, 是], [6, 3, 1, 1, 是], [6, 4, 0, 0, 是], [6, 5, 9, 9, 是], [6, 6, 8, 8, 是], [6, 7, 7, 7, 是], [6, 8, 6, 6, 是], [6, 9, 5, 5, 是], [7, 0, 3, 3, 是], [7, 1, 2, 2, 是], [7, 2, 1, 1, 是], [7, 3, 0, 0, 是], [7, 4, 9, 9, 是], [7, 5, 8, 8, 是], [7, 6, 7, 7, 是], [7, 7, 6, 6, 是], [7, 8, 5, 5, 是], [7, 9, 4, 4, 是], [8, 0, 2, 2, 是], [8, 1, 1, 1, 是], [8, 2, 0, 0, 是], [8, 3, 9, 9, 是], [8, 4, 8, 8, 是], [8, 5, 7, 7, 是], [8, 6, 6, 6, 是], [8, 7, 5, 5, 是], [8, 8, 4, 4, 是], [8, 9, 3, 3, 是], [9, 0, 1, 1, 是], [9, 1, 0, 0, 是], [9, 2, 9, 9, 是], [9, 3, 8, 8, 是], [9, 4, 7, 7, 是], [9, 5, 6, 6, 是], [9, 6, 5, 5, 是], [9, 7, 4, 4, 是], [9, 8, 3, 3, 是], [9, 9, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141008,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"target_identity":0,"mapping":[0,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0009","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],\"target_identity\":0,\"mapping\":[0,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 11, [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 10, 10, 是], [0, 2, 9, 9, 是], [0, 3, 8, 8, 是], [0, 4, 7, 7, 是], [0, 5, 6, 6, 是], [0, 6, 5, 5, 是], [0, 7, 4, 4, 是], [0, 8, 3, 3, 是], [0, 9, 2, 2, 是], [0, 10, 1, 1, 是], [1, 0, 10, 10, 是], [1, 1, 9, 9, 是], [1, 2, 8, 8, 是], [1, 3, 7, 7, 是], [1, 4, 6, 6, 是], [1, 5, 5, 5, 是], [1, 6, 4, 4, 是], [1, 7, 3, 3, 是], [1, 8, 2, 2, 是], [1, 9, 1, 1, 是], [1, 10, 0, 0, 是], [2, 0, 9, 9, 是], [2, 1, 8, 8, 是], [2, 2, 7, 7, 是], [2, 3, 6, 6, 是], [2, 4, 5, 5, 是], [2, 5, 4, 4, 是], [2, 6, 3, 3, 是], [2, 7, 2, 2, 是], [2, 8, 1, 1, 是], [2, 9, 0, 0, 是], [2, 10, 10, 10, 是], [3, 0, 8, 8, 是], [3, 1, 7, 7, 是], [3, 2, 6, 6, 是], [3, 3, 5, 5, 是], [3, 4, 4, 4, 是], [3, 5, 3, 3, 是], [3, 6, 2, 2, 是], [3, 7, 1, 1, 是], [3, 8, 0, 0, 是], [3, 9, 10, 10, 是], [3, 10, 9, 9, 是], [4, 0, 7, 7, 是], [4, 1, 6, 6, 是], [4, 2, 5, 5, 是], [4, 3, 4, 4, 是], [4, 4, 3, 3, 是], [4, 5, 2, 2, 是], [4, 6, 1, 1, 是], [4, 7, 0, 0, 是], [4, 8, 10, 10, 是], [4, 9, 9, 9, 是], [4, 10, 8, 8, 是], [5, 0, 6, 6, 是], [5, 1, 5, 5, 是], [5, 2, 4, 4, 是], [5, 3, 3, 3, 是], [5, 4, 2, 2, 是], [5, 5, 1, 1, 是], [5, 6, 0, 0, 是], [5, 7, 10, 10, 是], [5, 8, 9, 9, 是], [5, 9, 8, 8, 是], [5, 10, 7, 7, 是], [6, 0, 5, 5, 是], [6, 1, 4, 4, 是], [6, 2, 3, 3, 是], [6, 3, 2, 2, 是], [6, 4, 1, 1, 是], [6, 5, 0, 0, 是], [6, 6, 10, 10, 是], [6, 7, 9, 9, 是], [6, 8, 8, 8, 是], [6, 9, 7, 7, 是], [6, 10, 6, 6, 是], [7, 0, 4, 4, 是], [7, 1, 3, 3, 是], [7, 2, 2, 2, 是], [7, 3, 1, 1, 是], [7, 4, 0, 0, 是], [7, 5, 10, 10, 是], [7, 6, 9, 9, 是], [7, 7, 8, 8, 是], [7, 8, 7, 7, 是], [7, 9, 6, 6, 是], [7, 10, 5, 5, 是], [8, 0, 3, 3, 是], [8, 1, 2, 2, 是], [8, 2, 1, 1, 是], [8, 3, 0, 0, 是], [8, 4, 10, 10, 是], [8, 5, 9, 9, 是], [8, 6, 8, 8, 是], [8, 7, 7, 7, 是], [8, 8, 6, 6, 是], [8, 9, 5, 5, 是], [8, 10, 4, 4, 是], [9, 0, 2, 2, 是], [9, 1, 1, 1, 是], [9, 2, 0, 0, 是], [9, 3, 10, 10, 是], [9, 4, 9, 9, 是], [9, 5, 8, 8, 是], [9, 6, 7, 7, 是], [9, 7, 6, 6, 是], [9, 8, 5, 5, 是], [9, 9, 4, 4, 是], [9, 10, 3, 3, 是], [10, 0, 1, 1, 是], [10, 1, 0, 0, 是], [10, 2, 10, 10, 是], [10, 3, 9, 9, 是], [10, 4, 8, 8, 是], [10, 5, 7, 7, 是], [10, 6, 6, 6, 是], [10, 7, 5, 5, 是], [10, 8, 4, 4, 是], [10, 9, 3, 3, 是], [10, 10, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(11, 11, [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 10, 10, 是], [0, 2, 9, 9, 是], [0, 3, 8, 8, 是], [0, 4, 7, 7, 是], [0, 5, 6, 6, 是], [0, 6, 5, 5, 是], [0, 7, 4, 4, 是], [0, 8, 3, 3, 是], [0, 9, 2, 2, 是], [0, 10, 1, 1, 是], [1, 0, 10, 10, 是], [1, 1, 9, 9, 是], [1, 2, 8, 8, 是], [1, 3, 7, 7, 是], [1, 4, 6, 6, 是], [1, 5, 5, 5, 是], [1, 6, 4, 4, 是], [1, 7, 3, 3, 是], [1, 8, 2, 2, 是], [1, 9, 1, 1, 是], [1, 10, 0, 0, 是], [2, 0, 9, 9, 是], [2, 1, 8, 8, 是], [2, 2, 7, 7, 是], [2, 3, 6, 6, 是], [2, 4, 5, 5, 是], [2, 5, 4, 4, 是], [2, 6, 3, 3, 是], [2, 7, 2, 2, 是], [2, 8, 1, 1, 是], [2, 9, 0, 0, 是], [2, 10, 10, 10, 是], [3, 0, 8, 8, 是], [3, 1, 7, 7, 是], [3, 2, 6, 6, 是], [3, 3, 5, 5, 是], [3, 4, 4, 4, 是], [3, 5, 3, 3, 是], [3, 6, 2, 2, 是], [3, 7, 1, 1, 是], [3, 8, 0, 0, 是], [3, 9, 10, 10, 是], [3, 10, 9, 9, 是], [4, 0, 7, 7, 是], [4, 1, 6, 6, 是], [4, 2, 5, 5, 是], [4, 3, 4, 4, 是], [4, 4, 3, 3, 是], [4, 5, 2, 2, 是], [4, 6, 1, 1, 是], [4, 7, 0, 0, 是], [4, 8, 10, 10, 是], [4, 9, 9, 9, 是], [4, 10, 8, 8, 是], [5, 0, 6, 6, 是], [5, 1, 5, 5, 是], [5, 2, 4, 4, 是], [5, 3, 3, 3, 是], [5, 4, 2, 2, 是], [5, 5, 1, 1, 是], [5, 6, 0, 0, 是], [5, 7, 10, 10, 是], [5, 8, 9, 9, 是], [5, 9, 8, 8, 是], [5, 10, 7, 7, 是], [6, 0, 5, 5, 是], [6, 1, 4, 4, 是], [6, 2, 3, 3, 是], [6, 3, 2, 2, 是], [6, 4, 1, 1, 是], [6, 5, 0, 0, 是], [6, 6, 10, 10, 是], [6, 7, 9, 9, 是], [6, 8, 8, 8, 是], [6, 9, 7, 7, 是], [6, 10, 6, 6, 是], [7, 0, 4, 4, 是], [7, 1, 3, 3, 是], [7, 2, 2, 2, 是], [7, 3, 1, 1, 是], [7, 4, 0, 0, 是], [7, 5, 10, 10, 是], [7, 6, 9, 9, 是], [7, 7, 8, 8, 是], [7, 8, 7, 7, 是], [7, 9, 6, 6, 是], [7, 10, 5, 5, 是], [8, 0, 3, 3, 是], [8, 1, 2, 2, 是], [8, 2, 1, 1, 是], [8, 3, 0, 0, 是], [8, 4, 10, 10, 是], [8, 5, 9, 9, 是], [8, 6, 8, 8, 是], [8, 7, 7, 7, 是], [8, 8, 6, 6, 是], [8, 9, 5, 5, 是], [8, 10, 4, 4, 是], [9, 0, 2, 2, 是], [9, 1, 1, 1, 是], [9, 2, 0, 0, 是], [9, 3, 10, 10, 是], [9, 4, 9, 9, 是], [9, 5, 8, 8, 是], [9, 6, 7, 7, 是], [9, 7, 6, 6, 是], [9, 8, 5, 5, 是], [9, 9, 4, 4, 是], [9, 10, 3, 3, 是], [10, 0, 1, 1, 是], [10, 1, 0, 0, 是], [10, 2, 10, 10, 是], [10, 3, 9, 9, 是], [10, 4, 8, 8, 是], [10, 5, 7, 7, 是], [10, 6, 6, 6, 是], [10, 7, 5, 5, 是], [10, 8, 4, 4, 是], [10, 9, 3, 3, 是], [10, 10, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141009,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],"target_identity":0,"mapping":[0,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0010","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"mapping\":[0,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 12, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 11, 11, 是], [0, 2, 10, 10, 是], [0, 3, 9, 9, 是], [0, 4, 8, 8, 是], [0, 5, 7, 7, 是], [0, 6, 6, 6, 是], [0, 7, 5, 5, 是], [0, 8, 4, 4, 是], [0, 9, 3, 3, 是], [0, 10, 2, 2, 是], [0, 11, 1, 1, 是], [1, 0, 11, 11, 是], [1, 1, 10, 10, 是], [1, 2, 9, 9, 是], [1, 3, 8, 8, 是], [1, 4, 7, 7, 是], [1, 5, 6, 6, 是], [1, 6, 5, 5, 是], [1, 7, 4, 4, 是], [1, 8, 3, 3, 是], [1, 9, 2, 2, 是], [1, 10, 1, 1, 是], [1, 11, 0, 0, 是], [2, 0, 10, 10, 是], [2, 1, 9, 9, 是], [2, 2, 8, 8, 是], [2, 3, 7, 7, 是], [2, 4, 6, 6, 是], [2, 5, 5, 5, 是], [2, 6, 4, 4, 是], [2, 7, 3, 3, 是], [2, 8, 2, 2, 是], [2, 9, 1, 1, 是], [2, 10, 0, 0, 是], [2, 11, 11, 11, 是], [3, 0, 9, 9, 是], [3, 1, 8, 8, 是], [3, 2, 7, 7, 是], [3, 3, 6, 6, 是], [3, 4, 5, 5, 是], [3, 5, 4, 4, 是], [3, 6, 3, 3, 是], [3, 7, 2, 2, 是], [3, 8, 1, 1, 是], [3, 9, 0, 0, 是], [3, 10, 11, 11, 是], [3, 11, 10, 10, 是], [4, 0, 8, 8, 是], [4, 1, 7, 7, 是], [4, 2, 6, 6, 是], [4, 3, 5, 5, 是], [4, 4, 4, 4, 是], [4, 5, 3, 3, 是], [4, 6, 2, 2, 是], [4, 7, 1, 1, 是], [4, 8, 0, 0, 是], [4, 9, 11, 11, 是], [4, 10, 10, 10, 是], [4, 11, 9, 9, 是], [5, 0, 7, 7, 是], [5, 1, 6, 6, 是], [5, 2, 5, 5, 是], [5, 3, 4, 4, 是], [5, 4, 3, 3, 是], [5, 5, 2, 2, 是], [5, 6, 1, 1, 是], [5, 7, 0, 0, 是], [5, 8, 11, 11, 是], [5, 9, 10, 10, 是], [5, 10, 9, 9, 是], [5, 11, 8, 8, 是], [6, 0, 6, 6, 是], [6, 1, 5, 5, 是], [6, 2, 4, 4, 是], [6, 3, 3, 3, 是], [6, 4, 2, 2, 是], [6, 5, 1, 1, 是], [6, 6, 0, 0, 是], [6, 7, 11, 11, 是], [6, 8, 10, 10, 是], [6, 9, 9, 9, 是], [6, 10, 8, 8, 是], [6, 11, 7, 7, 是], [7, 0, 5, 5, 是], [7, 1, 4, 4, 是], [7, 2, 3, 3, 是], [7, 3, 2, 2, 是], [7, 4, 1, 1, 是], [7, 5, 0, 0, 是], [7, 6, 11, 11, 是], [7, 7, 10, 10, 是], [7, 8, 9, 9, 是], [7, 9, 8, 8, 是], [7, 10, 7, 7, 是], [7, 11, 6, 6, 是], [8, 0, 4, 4, 是], [8, 1, 3, 3, 是], [8, 2, 2, 2, 是], [8, 3, 1, 1, 是], [8, 4, 0, 0, 是], [8, 5, 11, 11, 是], [8, 6, 10, 10, 是], [8, 7, 9, 9, 是], [8, 8, 8, 8, 是], [8, 9, 7, 7, 是], [8, 10, 6, 6, 是], [8, 11, 5, 5, 是], [9, 0, 3, 3, 是], [9, 1, 2, 2, 是], [9, 2, 1, 1, 是], [9, 3, 0, 0, 是], [9, 4, 11, 11, 是], [9, 5, 10, 10, 是], [9, 6, 9, 9, 是], [9, 7, 8, 8, 是], [9, 8, 7, 7, 是], [9, 9, 6, 6, 是], [9, 10, 5, 5, 是], [9, 11, 4, 4, 是], [10, 0, 2, 2, 是], [10, 1, 1, 1, 是], [10, 2, 0, 0, 是], [10, 3, 11, 11, 是], [10, 4, 10, 10, 是], [10, 5, 9, 9, 是], [10, 6, 8, 8, 是], [10, 7, 7, 7, 是], [10, 8, 6, 6, 是], [10, 9, 5, 5, 是], [10, 10, 4, 4, 是], [10, 11, 3, 3, 是], [11, 0, 1, 1, 是], [11, 1, 0, 0, 是], [11, 2, 11, 11, 是], [11, 3, 10, 10, 是], [11, 4, 9, 9, 是], [11, 5, 8, 8, 是], [11, 6, 7, 7, 是], [11, 7, 6, 6, 是], [11, 8, 5, 5, 是], [11, 9, 4, 4, 是], [11, 10, 3, 3, 是], [11, 11, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(12, 12, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 11, 11, 是], [0, 2, 10, 10, 是], [0, 3, 9, 9, 是], [0, 4, 8, 8, 是], [0, 5, 7, 7, 是], [0, 6, 6, 6, 是], [0, 7, 5, 5, 是], [0, 8, 4, 4, 是], [0, 9, 3, 3, 是], [0, 10, 2, 2, 是], [0, 11, 1, 1, 是], [1, 0, 11, 11, 是], [1, 1, 10, 10, 是], [1, 2, 9, 9, 是], [1, 3, 8, 8, 是], [1, 4, 7, 7, 是], [1, 5, 6, 6, 是], [1, 6, 5, 5, 是], [1, 7, 4, 4, 是], [1, 8, 3, 3, 是], [1, 9, 2, 2, 是], [1, 10, 1, 1, 是], [1, 11, 0, 0, 是], [2, 0, 10, 10, 是], [2, 1, 9, 9, 是], [2, 2, 8, 8, 是], [2, 3, 7, 7, 是], [2, 4, 6, 6, 是], [2, 5, 5, 5, 是], [2, 6, 4, 4, 是], [2, 7, 3, 3, 是], [2, 8, 2, 2, 是], [2, 9, 1, 1, 是], [2, 10, 0, 0, 是], [2, 11, 11, 11, 是], [3, 0, 9, 9, 是], [3, 1, 8, 8, 是], [3, 2, 7, 7, 是], [3, 3, 6, 6, 是], [3, 4, 5, 5, 是], [3, 5, 4, 4, 是], [3, 6, 3, 3, 是], [3, 7, 2, 2, 是], [3, 8, 1, 1, 是], [3, 9, 0, 0, 是], [3, 10, 11, 11, 是], [3, 11, 10, 10, 是], [4, 0, 8, 8, 是], [4, 1, 7, 7, 是], [4, 2, 6, 6, 是], [4, 3, 5, 5, 是], [4, 4, 4, 4, 是], [4, 5, 3, 3, 是], [4, 6, 2, 2, 是], [4, 7, 1, 1, 是], [4, 8, 0, 0, 是], [4, 9, 11, 11, 是], [4, 10, 10, 10, 是], [4, 11, 9, 9, 是], [5, 0, 7, 7, 是], [5, 1, 6, 6, 是], [5, 2, 5, 5, 是], [5, 3, 4, 4, 是], [5, 4, 3, 3, 是], [5, 5, 2, 2, 是], [5, 6, 1, 1, 是], [5, 7, 0, 0, 是], [5, 8, 11, 11, 是], [5, 9, 10, 10, 是], [5, 10, 9, 9, 是], [5, 11, 8, 8, 是], [6, 0, 6, 6, 是], [6, 1, 5, 5, 是], [6, 2, 4, 4, 是], [6, 3, 3, 3, 是], [6, 4, 2, 2, 是], [6, 5, 1, 1, 是], [6, 6, 0, 0, 是], [6, 7, 11, 11, 是], [6, 8, 10, 10, 是], [6, 9, 9, 9, 是], [6, 10, 8, 8, 是], [6, 11, 7, 7, 是], [7, 0, 5, 5, 是], [7, 1, 4, 4, 是], [7, 2, 3, 3, 是], [7, 3, 2, 2, 是], [7, 4, 1, 1, 是], [7, 5, 0, 0, 是], [7, 6, 11, 11, 是], [7, 7, 10, 10, 是], [7, 8, 9, 9, 是], [7, 9, 8, 8, 是], [7, 10, 7, 7, 是], [7, 11, 6, 6, 是], [8, 0, 4, 4, 是], [8, 1, 3, 3, 是], [8, 2, 2, 2, 是], [8, 3, 1, 1, 是], [8, 4, 0, 0, 是], [8, 5, 11, 11, 是], [8, 6, 10, 10, 是], [8, 7, 9, 9, 是], [8, 8, 8, 8, 是], [8, 9, 7, 7, 是], [8, 10, 6, 6, 是], [8, 11, 5, 5, 是], [9, 0, 3, 3, 是], [9, 1, 2, 2, 是], [9, 2, 1, 1, 是], [9, 3, 0, 0, 是], [9, 4, 11, 11, 是], [9, 5, 10, 10, 是], [9, 6, 9, 9, 是], [9, 7, 8, 8, 是], [9, 8, 7, 7, 是], [9, 9, 6, 6, 是], [9, 10, 5, 5, 是], [9, 11, 4, 4, 是], [10, 0, 2, 2, 是], [10, 1, 1, 1, 是], [10, 2, 0, 0, 是], [10, 3, 11, 11, 是], [10, 4, 10, 10, 是], [10, 5, 9, 9, 是], [10, 6, 8, 8, 是], [10, 7, 7, 7, 是], [10, 8, 6, 6, 是], [10, 9, 5, 5, 是], [10, 10, 4, 4, 是], [10, 11, 3, 3, 是], [11, 0, 1, 1, 是], [11, 1, 0, 0, 是], [11, 2, 11, 11, 是], [11, 3, 10, 10, 是], [11, 4, 9, 9, 是], [11, 5, 8, 8, 是], [11, 6, 7, 7, 是], [11, 7, 6, 6, 是], [11, 8, 5, 5, 是], [11, 9, 4, 4, 是], [11, 10, 3, 3, 是], [11, 11, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141010,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"target_identity":0,"mapping":[0,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0011","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],\"target_identity\":0,\"mapping\":[0,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 13, [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 12, 12, 是], [0, 2, 11, 11, 是], [0, 3, 10, 10, 是], [0, 4, 9, 9, 是], [0, 5, 8, 8, 是], [0, 6, 7, 7, 是], [0, 7, 6, 6, 是], [0, 8, 5, 5, 是], [0, 9, 4, 4, 是], [0, 10, 3, 3, 是], [0, 11, 2, 2, 是], [0, 12, 1, 1, 是], [1, 0, 12, 12, 是], [1, 1, 11, 11, 是], [1, 2, 10, 10, 是], [1, 3, 9, 9, 是], [1, 4, 8, 8, 是], [1, 5, 7, 7, 是], [1, 6, 6, 6, 是], [1, 7, 5, 5, 是], [1, 8, 4, 4, 是], [1, 9, 3, 3, 是], [1, 10, 2, 2, 是], [1, 11, 1, 1, 是], [1, 12, 0, 0, 是], [2, 0, 11, 11, 是], [2, 1, 10, 10, 是], [2, 2, 9, 9, 是], [2, 3, 8, 8, 是], [2, 4, 7, 7, 是], [2, 5, 6, 6, 是], [2, 6, 5, 5, 是], [2, 7, 4, 4, 是], [2, 8, 3, 3, 是], [2, 9, 2, 2, 是], [2, 10, 1, 1, 是], [2, 11, 0, 0, 是], [2, 12, 12, 12, 是], [3, 0, 10, 10, 是], [3, 1, 9, 9, 是], [3, 2, 8, 8, 是], [3, 3, 7, 7, 是], [3, 4, 6, 6, 是], [3, 5, 5, 5, 是], [3, 6, 4, 4, 是], [3, 7, 3, 3, 是], [3, 8, 2, 2, 是], [3, 9, 1, 1, 是], [3, 10, 0, 0, 是], [3, 11, 12, 12, 是], [3, 12, 11, 11, 是], [4, 0, 9, 9, 是], [4, 1, 8, 8, 是], [4, 2, 7, 7, 是], [4, 3, 6, 6, 是], [4, 4, 5, 5, 是], [4, 5, 4, 4, 是], [4, 6, 3, 3, 是], [4, 7, 2, 2, 是], [4, 8, 1, 1, 是], [4, 9, 0, 0, 是], [4, 10, 12, 12, 是], [4, 11, 11, 11, 是], [4, 12, 10, 10, 是], [5, 0, 8, 8, 是], [5, 1, 7, 7, 是], [5, 2, 6, 6, 是], [5, 3, 5, 5, 是], [5, 4, 4, 4, 是], [5, 5, 3, 3, 是], [5, 6, 2, 2, 是], [5, 7, 1, 1, 是], [5, 8, 0, 0, 是], [5, 9, 12, 12, 是], [5, 10, 11, 11, 是], [5, 11, 10, 10, 是], [5, 12, 9, 9, 是], [6, 0, 7, 7, 是], [6, 1, 6, 6, 是], [6, 2, 5, 5, 是], [6, 3, 4, 4, 是], [6, 4, 3, 3, 是], [6, 5, 2, 2, 是], [6, 6, 1, 1, 是], [6, 7, 0, 0, 是], [6, 8, 12, 12, 是], [6, 9, 11, 11, 是], [6, 10, 10, 10, 是], [6, 11, 9, 9, 是], [6, 12, 8, 8, 是], [7, 0, 6, 6, 是], [7, 1, 5, 5, 是], [7, 2, 4, 4, 是], [7, 3, 3, 3, 是], [7, 4, 2, 2, 是], [7, 5, 1, 1, 是], [7, 6, 0, 0, 是], [7, 7, 12, 12, 是], [7, 8, 11, 11, 是], [7, 9, 10, 10, 是], [7, 10, 9, 9, 是], [7, 11, 8, 8, 是], [7, 12, 7, 7, 是], [8, 0, 5, 5, 是], [8, 1, 4, 4, 是], [8, 2, 3, 3, 是], [8, 3, 2, 2, 是], [8, 4, 1, 1, 是], [8, 5, 0, 0, 是], [8, 6, 12, 12, 是], [8, 7, 11, 11, 是], [8, 8, 10, 10, 是], [8, 9, 9, 9, 是], [8, 10, 8, 8, 是], [8, 11, 7, 7, 是], [8, 12, 6, 6, 是], [9, 0, 4, 4, 是], [9, 1, 3, 3, 是], [9, 2, 2, 2, 是], [9, 3, 1, 1, 是], [9, 4, 0, 0, 是], [9, 5, 12, 12, 是], [9, 6, 11, 11, 是], [9, 7, 10, 10, 是], [9, 8, 9, 9, 是], [9, 9, 8, 8, 是], [9, 10, 7, 7, 是], [9, 11, 6, 6, 是], [9, 12, 5, 5, 是], [10, 0, 3, 3, 是], [10, 1, 2, 2, 是], [10, 2, 1, 1, 是], [10, 3, 0, 0, 是], [10, 4, 12, 12, 是], [10, 5, 11, 11, 是], [10, 6, 10, 10, 是], [10, 7, 9, 9, 是], [10, 8, 8, 8, 是], [10, 9, 7, 7, 是], [10, 10, 6, 6, 是], [10, 11, 5, 5, 是], [10, 12, 4, 4, 是], [11, 0, 2, 2, 是], [11, 1, 1, 1, 是], [11, 2, 0, 0, 是], [11, 3, 12, 12, 是], [11, 4, 11, 11, 是], [11, 5, 10, 10, 是], [11, 6, 9, 9, 是], [11, 7, 8, 8, 是], [11, 8, 7, 7, 是], [11, 9, 6, 6, 是], [11, 10, 5, 5, 是], [11, 11, 4, 4, 是], [11, 12, 3, 3, 是], [12, 0, 1, 1, 是], [12, 1, 0, 0, 是], [12, 2, 12, 12, 是], [12, 3, 11, 11, 是], [12, 4, 10, 10, 是], [12, 5, 9, 9, 是], [12, 6, 8, 8, 是], [12, 7, 7, 7, 是], [12, 8, 6, 6, 是], [12, 9, 5, 5, 是], [12, 10, 4, 4, 是], [12, 11, 3, 3, 是], [12, 12, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(13, 13, [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 12, 12, 是], [0, 2, 11, 11, 是], [0, 3, 10, 10, 是], [0, 4, 9, 9, 是], [0, 5, 8, 8, 是], [0, 6, 7, 7, 是], [0, 7, 6, 6, 是], [0, 8, 5, 5, 是], [0, 9, 4, 4, 是], [0, 10, 3, 3, 是], [0, 11, 2, 2, 是], [0, 12, 1, 1, 是], [1, 0, 12, 12, 是], [1, 1, 11, 11, 是], [1, 2, 10, 10, 是], [1, 3, 9, 9, 是], [1, 4, 8, 8, 是], [1, 5, 7, 7, 是], [1, 6, 6, 6, 是], [1, 7, 5, 5, 是], [1, 8, 4, 4, 是], [1, 9, 3, 3, 是], [1, 10, 2, 2, 是], [1, 11, 1, 1, 是], [1, 12, 0, 0, 是], [2, 0, 11, 11, 是], [2, 1, 10, 10, 是], [2, 2, 9, 9, 是], [2, 3, 8, 8, 是], [2, 4, 7, 7, 是], [2, 5, 6, 6, 是], [2, 6, 5, 5, 是], [2, 7, 4, 4, 是], [2, 8, 3, 3, 是], [2, 9, 2, 2, 是], [2, 10, 1, 1, 是], [2, 11, 0, 0, 是], [2, 12, 12, 12, 是], [3, 0, 10, 10, 是], [3, 1, 9, 9, 是], [3, 2, 8, 8, 是], [3, 3, 7, 7, 是], [3, 4, 6, 6, 是], [3, 5, 5, 5, 是], [3, 6, 4, 4, 是], [3, 7, 3, 3, 是], [3, 8, 2, 2, 是], [3, 9, 1, 1, 是], [3, 10, 0, 0, 是], [3, 11, 12, 12, 是], [3, 12, 11, 11, 是], [4, 0, 9, 9, 是], [4, 1, 8, 8, 是], [4, 2, 7, 7, 是], [4, 3, 6, 6, 是], [4, 4, 5, 5, 是], [4, 5, 4, 4, 是], [4, 6, 3, 3, 是], [4, 7, 2, 2, 是], [4, 8, 1, 1, 是], [4, 9, 0, 0, 是], [4, 10, 12, 12, 是], [4, 11, 11, 11, 是], [4, 12, 10, 10, 是], [5, 0, 8, 8, 是], [5, 1, 7, 7, 是], [5, 2, 6, 6, 是], [5, 3, 5, 5, 是], [5, 4, 4, 4, 是], [5, 5, 3, 3, 是], [5, 6, 2, 2, 是], [5, 7, 1, 1, 是], [5, 8, 0, 0, 是], [5, 9, 12, 12, 是], [5, 10, 11, 11, 是], [5, 11, 10, 10, 是], [5, 12, 9, 9, 是], [6, 0, 7, 7, 是], [6, 1, 6, 6, 是], [6, 2, 5, 5, 是], [6, 3, 4, 4, 是], [6, 4, 3, 3, 是], [6, 5, 2, 2, 是], [6, 6, 1, 1, 是], [6, 7, 0, 0, 是], [6, 8, 12, 12, 是], [6, 9, 11, 11, 是], [6, 10, 10, 10, 是], [6, 11, 9, 9, 是], [6, 12, 8, 8, 是], [7, 0, 6, 6, 是], [7, 1, 5, 5, 是], [7, 2, 4, 4, 是], [7, 3, 3, 3, 是], [7, 4, 2, 2, 是], [7, 5, 1, 1, 是], [7, 6, 0, 0, 是], [7, 7, 12, 12, 是], [7, 8, 11, 11, 是], [7, 9, 10, 10, 是], [7, 10, 9, 9, 是], [7, 11, 8, 8, 是], [7, 12, 7, 7, 是], [8, 0, 5, 5, 是], [8, 1, 4, 4, 是], [8, 2, 3, 3, 是], [8, 3, 2, 2, 是], [8, 4, 1, 1, 是], [8, 5, 0, 0, 是], [8, 6, 12, 12, 是], [8, 7, 11, 11, 是], [8, 8, 10, 10, 是], [8, 9, 9, 9, 是], [8, 10, 8, 8, 是], [8, 11, 7, 7, 是], [8, 12, 6, 6, 是], [9, 0, 4, 4, 是], [9, 1, 3, 3, 是], [9, 2, 2, 2, 是], [9, 3, 1, 1, 是], [9, 4, 0, 0, 是], [9, 5, 12, 12, 是], [9, 6, 11, 11, 是], [9, 7, 10, 10, 是], [9, 8, 9, 9, 是], [9, 9, 8, 8, 是], [9, 10, 7, 7, 是], [9, 11, 6, 6, 是], [9, 12, 5, 5, 是], [10, 0, 3, 3, 是], [10, 1, 2, 2, 是], [10, 2, 1, 1, 是], [10, 3, 0, 0, 是], [10, 4, 12, 12, 是], [10, 5, 11, 11, 是], [10, 6, 10, 10, 是], [10, 7, 9, 9, 是], [10, 8, 8, 8, 是], [10, 9, 7, 7, 是], [10, 10, 6, 6, 是], [10, 11, 5, 5, 是], [10, 12, 4, 4, 是], [11, 0, 2, 2, 是], [11, 1, 1, 1, 是], [11, 2, 0, 0, 是], [11, 3, 12, 12, 是], [11, 4, 11, 11, 是], [11, 5, 10, 10, 是], [11, 6, 9, 9, 是], [11, 7, 8, 8, 是], [11, 8, 7, 7, 是], [11, 9, 6, 6, 是], [11, 10, 5, 5, 是], [11, 11, 4, 4, 是], [11, 12, 3, 3, 是], [12, 0, 1, 1, 是], [12, 1, 0, 0, 是], [12, 2, 12, 12, 是], [12, 3, 11, 11, 是], [12, 4, 10, 10, 是], [12, 5, 9, 9, 是], [12, 6, 8, 8, 是], [12, 7, 7, 7, 是], [12, 8, 6, 6, 是], [12, 9, 5, 5, 是], [12, 10, 4, 4, 是], [12, 11, 3, 3, 是], [12, 12, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141011,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],"target_identity":0,"mapping":[0,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0012","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"mapping\":[0,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 14, [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 13, 13, 是], [0, 2, 12, 12, 是], [0, 3, 11, 11, 是], [0, 4, 10, 10, 是], [0, 5, 9, 9, 是], [0, 6, 8, 8, 是], [0, 7, 7, 7, 是], [0, 8, 6, 6, 是], [0, 9, 5, 5, 是], [0, 10, 4, 4, 是], [0, 11, 3, 3, 是], [0, 12, 2, 2, 是], [0, 13, 1, 1, 是], [1, 0, 13, 13, 是], [1, 1, 12, 12, 是], [1, 2, 11, 11, 是], [1, 3, 10, 10, 是], [1, 4, 9, 9, 是], [1, 5, 8, 8, 是], [1, 6, 7, 7, 是], [1, 7, 6, 6, 是], [1, 8, 5, 5, 是], [1, 9, 4, 4, 是], [1, 10, 3, 3, 是], [1, 11, 2, 2, 是], [1, 12, 1, 1, 是], [1, 13, 0, 0, 是], [2, 0, 12, 12, 是], [2, 1, 11, 11, 是], [2, 2, 10, 10, 是], [2, 3, 9, 9, 是], [2, 4, 8, 8, 是], [2, 5, 7, 7, 是], [2, 6, 6, 6, 是], [2, 7, 5, 5, 是], [2, 8, 4, 4, 是], [2, 9, 3, 3, 是], [2, 10, 2, 2, 是], [2, 11, 1, 1, 是], [2, 12, 0, 0, 是], [2, 13, 13, 13, 是], [3, 0, 11, 11, 是], [3, 1, 10, 10, 是], [3, 2, 9, 9, 是], [3, 3, 8, 8, 是], [3, 4, 7, 7, 是], [3, 5, 6, 6, 是], [3, 6, 5, 5, 是], [3, 7, 4, 4, 是], [3, 8, 3, 3, 是], [3, 9, 2, 2, 是], [3, 10, 1, 1, 是], [3, 11, 0, 0, 是], [3, 12, 13, 13, 是], [3, 13, 12, 12, 是], [4, 0, 10, 10, 是], [4, 1, 9, 9, 是], [4, 2, 8, 8, 是], [4, 3, 7, 7, 是], [4, 4, 6, 6, 是], [4, 5, 5, 5, 是], [4, 6, 4, 4, 是], [4, 7, 3, 3, 是], [4, 8, 2, 2, 是], [4, 9, 1, 1, 是], [4, 10, 0, 0, 是], [4, 11, 13, 13, 是], [4, 12, 12, 12, 是], [4, 13, 11, 11, 是], [5, 0, 9, 9, 是], [5, 1, 8, 8, 是], [5, 2, 7, 7, 是], [5, 3, 6, 6, 是], [5, 4, 5, 5, 是], [5, 5, 4, 4, 是], [5, 6, 3, 3, 是], [5, 7, 2, 2, 是], [5, 8, 1, 1, 是], [5, 9, 0, 0, 是], [5, 10, 13, 13, 是], [5, 11, 12, 12, 是], [5, 12, 11, 11, 是], [5, 13, 10, 10, 是], [6, 0, 8, 8, 是], [6, 1, 7, 7, 是], [6, 2, 6, 6, 是], [6, 3, 5, 5, 是], [6, 4, 4, 4, 是], [6, 5, 3, 3, 是], [6, 6, 2, 2, 是], [6, 7, 1, 1, 是], [6, 8, 0, 0, 是], [6, 9, 13, 13, 是], [6, 10, 12, 12, 是], [6, 11, 11, 11, 是], [6, 12, 10, 10, 是], [6, 13, 9, 9, 是], [7, 0, 7, 7, 是], [7, 1, 6, 6, 是], [7, 2, 5, 5, 是], [7, 3, 4, 4, 是], [7, 4, 3, 3, 是], [7, 5, 2, 2, 是], [7, 6, 1, 1, 是], [7, 7, 0, 0, 是], [7, 8, 13, 13, 是], [7, 9, 12, 12, 是], [7, 10, 11, 11, 是], [7, 11, 10, 10, 是], [7, 12, 9, 9, 是], [7, 13, 8, 8, 是], [8, 0, 6, 6, 是], [8, 1, 5, 5, 是], [8, 2, 4, 4, 是], [8, 3, 3, 3, 是], [8, 4, 2, 2, 是], [8, 5, 1, 1, 是], [8, 6, 0, 0, 是], [8, 7, 13, 13, 是], [8, 8, 12, 12, 是], [8, 9, 11, 11, 是], [8, 10, 10, 10, 是], [8, 11, 9, 9, 是], [8, 12, 8, 8, 是], [8, 13, 7, 7, 是], [9, 0, 5, 5, 是], [9, 1, 4, 4, 是], [9, 2, 3, 3, 是], [9, 3, 2, 2, 是], [9, 4, 1, 1, 是], [9, 5, 0, 0, 是], [9, 6, 13, 13, 是], [9, 7, 12, 12, 是], [9, 8, 11, 11, 是], [9, 9, 10, 10, 是], [9, 10, 9, 9, 是], [9, 11, 8, 8, 是], [9, 12, 7, 7, 是], [9, 13, 6, 6, 是], [10, 0, 4, 4, 是], [10, 1, 3, 3, 是], [10, 2, 2, 2, 是], [10, 3, 1, 1, 是], [10, 4, 0, 0, 是], [10, 5, 13, 13, 是], [10, 6, 12, 12, 是], [10, 7, 11, 11, 是], [10, 8, 10, 10, 是], [10, 9, 9, 9, 是], [10, 10, 8, 8, 是], [10, 11, 7, 7, 是], [10, 12, 6, 6, 是], [10, 13, 5, 5, 是], [11, 0, 3, 3, 是], [11, 1, 2, 2, 是], [11, 2, 1, 1, 是], [11, 3, 0, 0, 是], [11, 4, 13, 13, 是], [11, 5, 12, 12, 是], [11, 6, 11, 11, 是], [11, 7, 10, 10, 是], [11, 8, 9, 9, 是], [11, 9, 8, 8, 是], [11, 10, 7, 7, 是], [11, 11, 6, 6, 是], [11, 12, 5, 5, 是], [11, 13, 4, 4, 是], [12, 0, 2, 2, 是], [12, 1, 1, 1, 是], [12, 2, 0, 0, 是], [12, 3, 13, 13, 是], [12, 4, 12, 12, 是], [12, 5, 11, 11, 是], [12, 6, 10, 10, 是], [12, 7, 9, 9, 是], [12, 8, 8, 8, 是], [12, 9, 7, 7, 是], [12, 10, 6, 6, 是], [12, 11, 5, 5, 是], [12, 12, 4, 4, 是], [12, 13, 3, 3, 是], [13, 0, 1, 1, 是], [13, 1, 0, 0, 是], [13, 2, 13, 13, 是], [13, 3, 12, 12, 是], [13, 4, 11, 11, 是], [13, 5, 10, 10, 是], [13, 6, 9, 9, 是], [13, 7, 8, 8, 是], [13, 8, 7, 7, 是], [13, 9, 6, 6, 是], [13, 10, 5, 5, 是], [13, 11, 4, 4, 是], [13, 12, 3, 3, 是], [13, 13, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(14, 14, [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 13, 13, 是], [0, 2, 12, 12, 是], [0, 3, 11, 11, 是], [0, 4, 10, 10, 是], [0, 5, 9, 9, 是], [0, 6, 8, 8, 是], [0, 7, 7, 7, 是], [0, 8, 6, 6, 是], [0, 9, 5, 5, 是], [0, 10, 4, 4, 是], [0, 11, 3, 3, 是], [0, 12, 2, 2, 是], [0, 13, 1, 1, 是], [1, 0, 13, 13, 是], [1, 1, 12, 12, 是], [1, 2, 11, 11, 是], [1, 3, 10, 10, 是], [1, 4, 9, 9, 是], [1, 5, 8, 8, 是], [1, 6, 7, 7, 是], [1, 7, 6, 6, 是], [1, 8, 5, 5, 是], [1, 9, 4, 4, 是], [1, 10, 3, 3, 是], [1, 11, 2, 2, 是], [1, 12, 1, 1, 是], [1, 13, 0, 0, 是], [2, 0, 12, 12, 是], [2, 1, 11, 11, 是], [2, 2, 10, 10, 是], [2, 3, 9, 9, 是], [2, 4, 8, 8, 是], [2, 5, 7, 7, 是], [2, 6, 6, 6, 是], [2, 7, 5, 5, 是], [2, 8, 4, 4, 是], [2, 9, 3, 3, 是], [2, 10, 2, 2, 是], [2, 11, 1, 1, 是], [2, 12, 0, 0, 是], [2, 13, 13, 13, 是], [3, 0, 11, 11, 是], [3, 1, 10, 10, 是], [3, 2, 9, 9, 是], [3, 3, 8, 8, 是], [3, 4, 7, 7, 是], [3, 5, 6, 6, 是], [3, 6, 5, 5, 是], [3, 7, 4, 4, 是], [3, 8, 3, 3, 是], [3, 9, 2, 2, 是], [3, 10, 1, 1, 是], [3, 11, 0, 0, 是], [3, 12, 13, 13, 是], [3, 13, 12, 12, 是], [4, 0, 10, 10, 是], [4, 1, 9, 9, 是], [4, 2, 8, 8, 是], [4, 3, 7, 7, 是], [4, 4, 6, 6, 是], [4, 5, 5, 5, 是], [4, 6, 4, 4, 是], [4, 7, 3, 3, 是], [4, 8, 2, 2, 是], [4, 9, 1, 1, 是], [4, 10, 0, 0, 是], [4, 11, 13, 13, 是], [4, 12, 12, 12, 是], [4, 13, 11, 11, 是], [5, 0, 9, 9, 是], [5, 1, 8, 8, 是], [5, 2, 7, 7, 是], [5, 3, 6, 6, 是], [5, 4, 5, 5, 是], [5, 5, 4, 4, 是], [5, 6, 3, 3, 是], [5, 7, 2, 2, 是], [5, 8, 1, 1, 是], [5, 9, 0, 0, 是], [5, 10, 13, 13, 是], [5, 11, 12, 12, 是], [5, 12, 11, 11, 是], [5, 13, 10, 10, 是], [6, 0, 8, 8, 是], [6, 1, 7, 7, 是], [6, 2, 6, 6, 是], [6, 3, 5, 5, 是], [6, 4, 4, 4, 是], [6, 5, 3, 3, 是], [6, 6, 2, 2, 是], [6, 7, 1, 1, 是], [6, 8, 0, 0, 是], [6, 9, 13, 13, 是], [6, 10, 12, 12, 是], [6, 11, 11, 11, 是], [6, 12, 10, 10, 是], [6, 13, 9, 9, 是], [7, 0, 7, 7, 是], [7, 1, 6, 6, 是], [7, 2, 5, 5, 是], [7, 3, 4, 4, 是], [7, 4, 3, 3, 是], [7, 5, 2, 2, 是], [7, 6, 1, 1, 是], [7, 7, 0, 0, 是], [7, 8, 13, 13, 是], [7, 9, 12, 12, 是], [7, 10, 11, 11, 是], [7, 11, 10, 10, 是], [7, 12, 9, 9, 是], [7, 13, 8, 8, 是], [8, 0, 6, 6, 是], [8, 1, 5, 5, 是], [8, 2, 4, 4, 是], [8, 3, 3, 3, 是], [8, 4, 2, 2, 是], [8, 5, 1, 1, 是], [8, 6, 0, 0, 是], [8, 7, 13, 13, 是], [8, 8, 12, 12, 是], [8, 9, 11, 11, 是], [8, 10, 10, 10, 是], [8, 11, 9, 9, 是], [8, 12, 8, 8, 是], [8, 13, 7, 7, 是], [9, 0, 5, 5, 是], [9, 1, 4, 4, 是], [9, 2, 3, 3, 是], [9, 3, 2, 2, 是], [9, 4, 1, 1, 是], [9, 5, 0, 0, 是], [9, 6, 13, 13, 是], [9, 7, 12, 12, 是], [9, 8, 11, 11, 是], [9, 9, 10, 10, 是], [9, 10, 9, 9, 是], [9, 11, 8, 8, 是], [9, 12, 7, 7, 是], [9, 13, 6, 6, 是], [10, 0, 4, 4, 是], [10, 1, 3, 3, 是], [10, 2, 2, 2, 是], [10, 3, 1, 1, 是], [10, 4, 0, 0, 是], [10, 5, 13, 13, 是], [10, 6, 12, 12, 是], [10, 7, 11, 11, 是], [10, 8, 10, 10, 是], [10, 9, 9, 9, 是], [10, 10, 8, 8, 是], [10, 11, 7, 7, 是], [10, 12, 6, 6, 是], [10, 13, 5, 5, 是], [11, 0, 3, 3, 是], [11, 1, 2, 2, 是], [11, 2, 1, 1, 是], [11, 3, 0, 0, 是], [11, 4, 13, 13, 是], [11, 5, 12, 12, 是], [11, 6, 11, 11, 是], [11, 7, 10, 10, 是], [11, 8, 9, 9, 是], [11, 9, 8, 8, 是], [11, 10, 7, 7, 是], [11, 11, 6, 6, 是], [11, 12, 5, 5, 是], [11, 13, 4, 4, 是], [12, 0, 2, 2, 是], [12, 1, 1, 1, 是], [12, 2, 0, 0, 是], [12, 3, 13, 13, 是], [12, 4, 12, 12, 是], [12, 5, 11, 11, 是], [12, 6, 10, 10, 是], [12, 7, 9, 9, 是], [12, 8, 8, 8, 是], [12, 9, 7, 7, 是], [12, 10, 6, 6, 是], [12, 11, 5, 5, 是], [12, 12, 4, 4, 是], [12, 13, 3, 3, 是], [13, 0, 1, 1, 是], [13, 1, 0, 0, 是], [13, 2, 13, 13, 是], [13, 3, 12, 12, 是], [13, 4, 11, 11, 是], [13, 5, 10, 10, 是], [13, 6, 9, 9, 是], [13, 7, 8, 8, 是], [13, 8, 7, 7, 是], [13, 9, 6, 6, 是], [13, 10, 5, 5, 是], [13, 11, 4, 4, 是], [13, 12, 3, 3, 是], [13, 13, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141012,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"target_identity":0,"mapping":[0,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0013","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"mapping\":[0,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 14, 14, 是], [0, 2, 13, 13, 是], [0, 3, 12, 12, 是], [0, 4, 11, 11, 是], [0, 5, 10, 10, 是], [0, 6, 9, 9, 是], [0, 7, 8, 8, 是], [0, 8, 7, 7, 是], [0, 9, 6, 6, 是], [0, 10, 5, 5, 是], [0, 11, 4, 4, 是], [0, 12, 3, 3, 是], [0, 13, 2, 2, 是], [0, 14, 1, 1, 是], [1, 0, 14, 14, 是], [1, 1, 13, 13, 是], [1, 2, 12, 12, 是], [1, 3, 11, 11, 是], [1, 4, 10, 10, 是], [1, 5, 9, 9, 是], [1, 6, 8, 8, 是], [1, 7, 7, 7, 是], [1, 8, 6, 6, 是], [1, 9, 5, 5, 是], [1, 10, 4, 4, 是], [1, 11, 3, 3, 是], [1, 12, 2, 2, 是], [1, 13, 1, 1, 是], [1, 14, 0, 0, 是], [2, 0, 13, 13, 是], [2, 1, 12, 12, 是], [2, 2, 11, 11, 是], [2, 3, 10, 10, 是], [2, 4, 9, 9, 是], [2, 5, 8, 8, 是], [2, 6, 7, 7, 是], [2, 7, 6, 6, 是], [2, 8, 5, 5, 是], [2, 9, 4, 4, 是], [2, 10, 3, 3, 是], [2, 11, 2, 2, 是], [2, 12, 1, 1, 是], [2, 13, 0, 0, 是], [2, 14, 14, 14, 是], [3, 0, 12, 12, 是], [3, 1, 11, 11, 是], [3, 2, 10, 10, 是], [3, 3, 9, 9, 是], [3, 4, 8, 8, 是], [3, 5, 7, 7, 是], [3, 6, 6, 6, 是], [3, 7, 5, 5, 是], [3, 8, 4, 4, 是], [3, 9, 3, 3, 是], [3, 10, 2, 2, 是], [3, 11, 1, 1, 是], [3, 12, 0, 0, 是], [3, 13, 14, 14, 是], [3, 14, 13, 13, 是], [4, 0, 11, 11, 是], [4, 1, 10, 10, 是], [4, 2, 9, 9, 是], [4, 3, 8, 8, 是], [4, 4, 7, 7, 是], [4, 5, 6, 6, 是], [4, 6, 5, 5, 是], [4, 7, 4, 4, 是], [4, 8, 3, 3, 是], [4, 9, 2, 2, 是], [4, 10, 1, 1, 是], [4, 11, 0, 0, 是], [4, 12, 14, 14, 是], [4, 13, 13, 13, 是], [4, 14, 12, 12, 是], [5, 0, 10, 10, 是], [5, 1, 9, 9, 是], [5, 2, 8, 8, 是], [5, 3, 7, 7, 是], [5, 4, 6, 6, 是], [5, 5, 5, 5, 是], [5, 6, 4, 4, 是], [5, 7, 3, 3, 是], [5, 8, 2, 2, 是], [5, 9, 1, 1, 是], [5, 10, 0, 0, 是], [5, 11, 14, 14, 是], [5, 12, 13, 13, 是], [5, 13, 12, 12, 是], [5, 14, 11, 11, 是], [6, 0, 9, 9, 是], [6, 1, 8, 8, 是], [6, 2, 7, 7, 是], [6, 3, 6, 6, 是], [6, 4, 5, 5, 是], [6, 5, 4, 4, 是], [6, 6, 3, 3, 是], [6, 7, 2, 2, 是], [6, 8, 1, 1, 是], [6, 9, 0, 0, 是], [6, 10, 14, 14, 是], [6, 11, 13, 13, 是], [6, 12, 12, 12, 是], [6, 13, 11, 11, 是], [6, 14, 10, 10, 是], [7, 0, 8, 8, 是], [7, 1, 7, 7, 是], [7, 2, 6, 6, 是], [7, 3, 5, 5, 是], [7, 4, 4, 4, 是], [7, 5, 3, 3, 是], [7, 6, 2, 2, 是], [7, 7, 1, 1, 是], [7, 8, 0, 0, 是], [7, 9, 14, 14, 是], [7, 10, 13, 13, 是], [7, 11, 12, 12, 是], [7, 12, 11, 11, 是], [7, 13, 10, 10, 是], [7, 14, 9, 9, 是], [8, 0, 7, 7, 是], [8, 1, 6, 6, 是], [8, 2, 5, 5, 是], [8, 3, 4, 4, 是], [8, 4, 3, 3, 是], [8, 5, 2, 2, 是], [8, 6, 1, 1, 是], [8, 7, 0, 0, 是], [8, 8, 14, 14, 是], [8, 9, 13, 13, 是], [8, 10, 12, 12, 是], [8, 11, 11, 11, 是], [8, 12, 10, 10, 是], [8, 13, 9, 9, 是], [8, 14, 8, 8, 是], [9, 0, 6, 6, 是], [9, 1, 5, 5, 是], [9, 2, 4, 4, 是], [9, 3, 3, 3, 是], [9, 4, 2, 2, 是], [9, 5, 1, 1, 是], [9, 6, 0, 0, 是], [9, 7, 14, 14, 是], [9, 8, 13, 13, 是], [9, 9, 12, 12, 是], [9, 10, 11, 11, 是], [9, 11, 10, 10, 是], [9, 12, 9, 9, 是], [9, 13, 8, 8, 是], [9, 14, 7, 7, 是], [10, 0, 5, 5, 是], [10, 1, 4, 4, 是], [10, 2, 3, 3, 是], [10, 3, 2, 2, 是], [10, 4, 1, 1, 是], [10, 5, 0, 0, 是], [10, 6, 14, 14, 是], [10, 7, 13, 13, 是], [10, 8, 12, 12, 是], [10, 9, 11, 11, 是], [10, 10, 10, 10, 是], [10, 11, 9, 9, 是], [10, 12, 8, 8, 是], [10, 13, 7, 7, 是], [10, 14, 6, 6, 是], [11, 0, 4, 4, 是], [11, 1, 3, 3, 是], [11, 2, 2, 2, 是], [11, 3, 1, 1, 是], [11, 4, 0, 0, 是], [11, 5, 14, 14, 是], [11, 6, 13, 13, 是], [11, 7, 12, 12, 是], [11, 8, 11, 11, 是], [11, 9, 10, 10, 是], [11, 10, 9, 9, 是], [11, 11, 8, 8, 是], [11, 12, 7, 7, 是], [11, 13, 6, 6, 是], [11, 14, 5, 5, 是], [12, 0, 3, 3, 是], [12, 1, 2, 2, 是], [12, 2, 1, 1, 是], [12, 3, 0, 0, 是], [12, 4, 14, 14, 是], [12, 5, 13, 13, 是], [12, 6, 12, 12, 是], [12, 7, 11, 11, 是], [12, 8, 10, 10, 是], [12, 9, 9, 9, 是], [12, 10, 8, 8, 是], [12, 11, 7, 7, 是], [12, 12, 6, 6, 是], [12, 13, 5, 5, 是], [12, 14, 4, 4, 是], [13, 0, 2, 2, 是], [13, 1, 1, 1, 是], [13, 2, 0, 0, 是], [13, 3, 14, 14, 是], [13, 4, 13, 13, 是], [13, 5, 12, 12, 是], [13, 6, 11, 11, 是], [13, 7, 10, 10, 是], [13, 8, 9, 9, 是], [13, 9, 8, 8, 是], [13, 10, 7, 7, 是], [13, 11, 6, 6, 是], [13, 12, 5, 5, 是], [13, 13, 4, 4, 是], [13, 14, 3, 3, 是], [14, 0, 1, 1, 是], [14, 1, 0, 0, 是], [14, 2, 14, 14, 是], [14, 3, 13, 13, 是], [14, 4, 12, 12, 是], [14, 5, 11, 11, 是], [14, 6, 10, 10, 是], [14, 7, 9, 9, 是], [14, 8, 8, 8, 是], [14, 9, 7, 7, 是], [14, 10, 6, 6, 是], [14, 11, 5, 5, 是], [14, 12, 4, 4, 是], [14, 13, 3, 3, 是], [14, 14, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 14, 14, 是], [0, 2, 13, 13, 是], [0, 3, 12, 12, 是], [0, 4, 11, 11, 是], [0, 5, 10, 10, 是], [0, 6, 9, 9, 是], [0, 7, 8, 8, 是], [0, 8, 7, 7, 是], [0, 9, 6, 6, 是], [0, 10, 5, 5, 是], [0, 11, 4, 4, 是], [0, 12, 3, 3, 是], [0, 13, 2, 2, 是], [0, 14, 1, 1, 是], [1, 0, 14, 14, 是], [1, 1, 13, 13, 是], [1, 2, 12, 12, 是], [1, 3, 11, 11, 是], [1, 4, 10, 10, 是], [1, 5, 9, 9, 是], [1, 6, 8, 8, 是], [1, 7, 7, 7, 是], [1, 8, 6, 6, 是], [1, 9, 5, 5, 是], [1, 10, 4, 4, 是], [1, 11, 3, 3, 是], [1, 12, 2, 2, 是], [1, 13, 1, 1, 是], [1, 14, 0, 0, 是], [2, 0, 13, 13, 是], [2, 1, 12, 12, 是], [2, 2, 11, 11, 是], [2, 3, 10, 10, 是], [2, 4, 9, 9, 是], [2, 5, 8, 8, 是], [2, 6, 7, 7, 是], [2, 7, 6, 6, 是], [2, 8, 5, 5, 是], [2, 9, 4, 4, 是], [2, 10, 3, 3, 是], [2, 11, 2, 2, 是], [2, 12, 1, 1, 是], [2, 13, 0, 0, 是], [2, 14, 14, 14, 是], [3, 0, 12, 12, 是], [3, 1, 11, 11, 是], [3, 2, 10, 10, 是], [3, 3, 9, 9, 是], [3, 4, 8, 8, 是], [3, 5, 7, 7, 是], [3, 6, 6, 6, 是], [3, 7, 5, 5, 是], [3, 8, 4, 4, 是], [3, 9, 3, 3, 是], [3, 10, 2, 2, 是], [3, 11, 1, 1, 是], [3, 12, 0, 0, 是], [3, 13, 14, 14, 是], [3, 14, 13, 13, 是], [4, 0, 11, 11, 是], [4, 1, 10, 10, 是], [4, 2, 9, 9, 是], [4, 3, 8, 8, 是], [4, 4, 7, 7, 是], [4, 5, 6, 6, 是], [4, 6, 5, 5, 是], [4, 7, 4, 4, 是], [4, 8, 3, 3, 是], [4, 9, 2, 2, 是], [4, 10, 1, 1, 是], [4, 11, 0, 0, 是], [4, 12, 14, 14, 是], [4, 13, 13, 13, 是], [4, 14, 12, 12, 是], [5, 0, 10, 10, 是], [5, 1, 9, 9, 是], [5, 2, 8, 8, 是], [5, 3, 7, 7, 是], [5, 4, 6, 6, 是], [5, 5, 5, 5, 是], [5, 6, 4, 4, 是], [5, 7, 3, 3, 是], [5, 8, 2, 2, 是], [5, 9, 1, 1, 是], [5, 10, 0, 0, 是], [5, 11, 14, 14, 是], [5, 12, 13, 13, 是], [5, 13, 12, 12, 是], [5, 14, 11, 11, 是], [6, 0, 9, 9, 是], [6, 1, 8, 8, 是], [6, 2, 7, 7, 是], [6, 3, 6, 6, 是], [6, 4, 5, 5, 是], [6, 5, 4, 4, 是], [6, 6, 3, 3, 是], [6, 7, 2, 2, 是], [6, 8, 1, 1, 是], [6, 9, 0, 0, 是], [6, 10, 14, 14, 是], [6, 11, 13, 13, 是], [6, 12, 12, 12, 是], [6, 13, 11, 11, 是], [6, 14, 10, 10, 是], [7, 0, 8, 8, 是], [7, 1, 7, 7, 是], [7, 2, 6, 6, 是], [7, 3, 5, 5, 是], [7, 4, 4, 4, 是], [7, 5, 3, 3, 是], [7, 6, 2, 2, 是], [7, 7, 1, 1, 是], [7, 8, 0, 0, 是], [7, 9, 14, 14, 是], [7, 10, 13, 13, 是], [7, 11, 12, 12, 是], [7, 12, 11, 11, 是], [7, 13, 10, 10, 是], [7, 14, 9, 9, 是], [8, 0, 7, 7, 是], [8, 1, 6, 6, 是], [8, 2, 5, 5, 是], [8, 3, 4, 4, 是], [8, 4, 3, 3, 是], [8, 5, 2, 2, 是], [8, 6, 1, 1, 是], [8, 7, 0, 0, 是], [8, 8, 14, 14, 是], [8, 9, 13, 13, 是], [8, 10, 12, 12, 是], [8, 11, 11, 11, 是], [8, 12, 10, 10, 是], [8, 13, 9, 9, 是], [8, 14, 8, 8, 是], [9, 0, 6, 6, 是], [9, 1, 5, 5, 是], [9, 2, 4, 4, 是], [9, 3, 3, 3, 是], [9, 4, 2, 2, 是], [9, 5, 1, 1, 是], [9, 6, 0, 0, 是], [9, 7, 14, 14, 是], [9, 8, 13, 13, 是], [9, 9, 12, 12, 是], [9, 10, 11, 11, 是], [9, 11, 10, 10, 是], [9, 12, 9, 9, 是], [9, 13, 8, 8, 是], [9, 14, 7, 7, 是], [10, 0, 5, 5, 是], [10, 1, 4, 4, 是], [10, 2, 3, 3, 是], [10, 3, 2, 2, 是], [10, 4, 1, 1, 是], [10, 5, 0, 0, 是], [10, 6, 14, 14, 是], [10, 7, 13, 13, 是], [10, 8, 12, 12, 是], [10, 9, 11, 11, 是], [10, 10, 10, 10, 是], [10, 11, 9, 9, 是], [10, 12, 8, 8, 是], [10, 13, 7, 7, 是], [10, 14, 6, 6, 是], [11, 0, 4, 4, 是], [11, 1, 3, 3, 是], [11, 2, 2, 2, 是], [11, 3, 1, 1, 是], [11, 4, 0, 0, 是], [11, 5, 14, 14, 是], [11, 6, 13, 13, 是], [11, 7, 12, 12, 是], [11, 8, 11, 11, 是], [11, 9, 10, 10, 是], [11, 10, 9, 9, 是], [11, 11, 8, 8, 是], [11, 12, 7, 7, 是], [11, 13, 6, 6, 是], [11, 14, 5, 5, 是], [12, 0, 3, 3, 是], [12, 1, 2, 2, 是], [12, 2, 1, 1, 是], [12, 3, 0, 0, 是], [12, 4, 14, 14, 是], [12, 5, 13, 13, 是], [12, 6, 12, 12, 是], [12, 7, 11, 11, 是], [12, 8, 10, 10, 是], [12, 9, 9, 9, 是], [12, 10, 8, 8, 是], [12, 11, 7, 7, 是], [12, 12, 6, 6, 是], [12, 13, 5, 5, 是], [12, 14, 4, 4, 是], [13, 0, 2, 2, 是], [13, 1, 1, 1, 是], [13, 2, 0, 0, 是], [13, 3, 14, 14, 是], [13, 4, 13, 13, 是], [13, 5, 12, 12, 是], [13, 6, 11, 11, 是], [13, 7, 10, 10, 是], [13, 8, 9, 9, 是], [13, 9, 8, 8, 是], [13, 10, 7, 7, 是], [13, 11, 6, 6, 是], [13, 12, 5, 5, 是], [13, 13, 4, 4, 是], [13, 14, 3, 3, 是], [14, 0, 1, 1, 是], [14, 1, 0, 0, 是], [14, 2, 14, 14, 是], [14, 3, 13, 13, 是], [14, 4, 12, 12, 是], [14, 5, 11, 11, 是], [14, 6, 10, 10, 是], [14, 7, 9, 9, 是], [14, 8, 8, 8, 是], [14, 9, 7, 7, 是], [14, 10, 6, 6, 是], [14, 11, 5, 5, 是], [14, 12, 4, 4, 是], [14, 13, 3, 3, 是], [14, 14, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141013,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],"target_identity":0,"mapping":[0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0014","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"mapping\":[0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 16, [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 15, 15, 是], [0, 2, 14, 14, 是], [0, 3, 13, 13, 是], [0, 4, 12, 12, 是], [0, 5, 11, 11, 是], [0, 6, 10, 10, 是], [0, 7, 9, 9, 是], [0, 8, 8, 8, 是], [0, 9, 7, 7, 是], [0, 10, 6, 6, 是], [0, 11, 5, 5, 是], [0, 12, 4, 4, 是], [0, 13, 3, 3, 是], [0, 14, 2, 2, 是], [0, 15, 1, 1, 是], [1, 0, 15, 15, 是], [1, 1, 14, 14, 是], [1, 2, 13, 13, 是], [1, 3, 12, 12, 是], [1, 4, 11, 11, 是], [1, 5, 10, 10, 是], [1, 6, 9, 9, 是], [1, 7, 8, 8, 是], [1, 8, 7, 7, 是], [1, 9, 6, 6, 是], [1, 10, 5, 5, 是], [1, 11, 4, 4, 是], [1, 12, 3, 3, 是], [1, 13, 2, 2, 是], [1, 14, 1, 1, 是], [1, 15, 0, 0, 是], [2, 0, 14, 14, 是], [2, 1, 13, 13, 是], [2, 2, 12, 12, 是], [2, 3, 11, 11, 是], [2, 4, 10, 10, 是], [2, 5, 9, 9, 是], [2, 6, 8, 8, 是], [2, 7, 7, 7, 是], [2, 8, 6, 6, 是], [2, 9, 5, 5, 是], [2, 10, 4, 4, 是], [2, 11, 3, 3, 是], [2, 12, 2, 2, 是], [2, 13, 1, 1, 是], [2, 14, 0, 0, 是], [2, 15, 15, 15, 是], [3, 0, 13, 13, 是], [3, 1, 12, 12, 是], [3, 2, 11, 11, 是], [3, 3, 10, 10, 是], [3, 4, 9, 9, 是], [3, 5, 8, 8, 是], [3, 6, 7, 7, 是], [3, 7, 6, 6, 是], [3, 8, 5, 5, 是], [3, 9, 4, 4, 是], [3, 10, 3, 3, 是], [3, 11, 2, 2, 是], [3, 12, 1, 1, 是], [3, 13, 0, 0, 是], [3, 14, 15, 15, 是], [3, 15, 14, 14, 是], [4, 0, 12, 12, 是], [4, 1, 11, 11, 是], [4, 2, 10, 10, 是], [4, 3, 9, 9, 是], [4, 4, 8, 8, 是], [4, 5, 7, 7, 是], [4, 6, 6, 6, 是], [4, 7, 5, 5, 是], [4, 8, 4, 4, 是], [4, 9, 3, 3, 是], [4, 10, 2, 2, 是], [4, 11, 1, 1, 是], [4, 12, 0, 0, 是], [4, 13, 15, 15, 是], [4, 14, 14, 14, 是], [4, 15, 13, 13, 是], [5, 0, 11, 11, 是], [5, 1, 10, 10, 是], [5, 2, 9, 9, 是], [5, 3, 8, 8, 是], [5, 4, 7, 7, 是], [5, 5, 6, 6, 是], [5, 6, 5, 5, 是], [5, 7, 4, 4, 是], [5, 8, 3, 3, 是], [5, 9, 2, 2, 是], [5, 10, 1, 1, 是], [5, 11, 0, 0, 是], [5, 12, 15, 15, 是], [5, 13, 14, 14, 是], [5, 14, 13, 13, 是], [5, 15, 12, 12, 是], [6, 0, 10, 10, 是], [6, 1, 9, 9, 是], [6, 2, 8, 8, 是], [6, 3, 7, 7, 是], [6, 4, 6, 6, 是], [6, 5, 5, 5, 是], [6, 6, 4, 4, 是], [6, 7, 3, 3, 是], [6, 8, 2, 2, 是], [6, 9, 1, 1, 是], [6, 10, 0, 0, 是], [6, 11, 15, 15, 是], [6, 12, 14, 14, 是], [6, 13, 13, 13, 是], [6, 14, 12, 12, 是], [6, 15, 11, 11, 是], [7, 0, 9, 9, 是], [7, 1, 8, 8, 是], [7, 2, 7, 7, 是], [7, 3, 6, 6, 是], [7, 4, 5, 5, 是], [7, 5, 4, 4, 是], [7, 6, 3, 3, 是], [7, 7, 2, 2, 是], [7, 8, 1, 1, 是], [7, 9, 0, 0, 是], [7, 10, 15, 15, 是], [7, 11, 14, 14, 是], [7, 12, 13, 13, 是], [7, 13, 12, 12, 是], [7, 14, 11, 11, 是], [7, 15, 10, 10, 是], [8, 0, 8, 8, 是], [8, 1, 7, 7, 是], [8, 2, 6, 6, 是], [8, 3, 5, 5, 是], [8, 4, 4, 4, 是], [8, 5, 3, 3, 是], [8, 6, 2, 2, 是], [8, 7, 1, 1, 是], [8, 8, 0, 0, 是], [8, 9, 15, 15, 是], [8, 10, 14, 14, 是], [8, 11, 13, 13, 是], [8, 12, 12, 12, 是], [8, 13, 11, 11, 是], [8, 14, 10, 10, 是], [8, 15, 9, 9, 是], [9, 0, 7, 7, 是], [9, 1, 6, 6, 是], [9, 2, 5, 5, 是], [9, 3, 4, 4, 是], [9, 4, 3, 3, 是], [9, 5, 2, 2, 是], [9, 6, 1, 1, 是], [9, 7, 0, 0, 是], [9, 8, 15, 15, 是], [9, 9, 14, 14, 是], [9, 10, 13, 13, 是], [9, 11, 12, 12, 是], [9, 12, 11, 11, 是], [9, 13, 10, 10, 是], [9, 14, 9, 9, 是], [9, 15, 8, 8, 是], [10, 0, 6, 6, 是], [10, 1, 5, 5, 是], [10, 2, 4, 4, 是], [10, 3, 3, 3, 是], [10, 4, 2, 2, 是], [10, 5, 1, 1, 是], [10, 6, 0, 0, 是], [10, 7, 15, 15, 是], [10, 8, 14, 14, 是], [10, 9, 13, 13, 是], [10, 10, 12, 12, 是], [10, 11, 11, 11, 是], [10, 12, 10, 10, 是], [10, 13, 9, 9, 是], [10, 14, 8, 8, 是], [10, 15, 7, 7, 是], [11, 0, 5, 5, 是], [11, 1, 4, 4, 是], [11, 2, 3, 3, 是], [11, 3, 2, 2, 是], [11, 4, 1, 1, 是], [11, 5, 0, 0, 是], [11, 6, 15, 15, 是], [11, 7, 14, 14, 是], [11, 8, 13, 13, 是], [11, 9, 12, 12, 是], [11, 10, 11, 11, 是], [11, 11, 10, 10, 是], [11, 12, 9, 9, 是], [11, 13, 8, 8, 是], [11, 14, 7, 7, 是], [11, 15, 6, 6, 是], [12, 0, 4, 4, 是], [12, 1, 3, 3, 是], [12, 2, 2, 2, 是], [12, 3, 1, 1, 是], [12, 4, 0, 0, 是], [12, 5, 15, 15, 是], [12, 6, 14, 14, 是], [12, 7, 13, 13, 是], [12, 8, 12, 12, 是], [12, 9, 11, 11, 是], [12, 10, 10, 10, 是], [12, 11, 9, 9, 是], [12, 12, 8, 8, 是], [12, 13, 7, 7, 是], [12, 14, 6, 6, 是], [12, 15, 5, 5, 是], [13, 0, 3, 3, 是], [13, 1, 2, 2, 是], [13, 2, 1, 1, 是], [13, 3, 0, 0, 是], [13, 4, 15, 15, 是], [13, 5, 14, 14, 是], [13, 6, 13, 13, 是], [13, 7, 12, 12, 是], [13, 8, 11, 11, 是], [13, 9, 10, 10, 是], [13, 10, 9, 9, 是], [13, 11, 8, 8, 是], [13, 12, 7, 7, 是], [13, 13, 6, 6, 是], [13, 14, 5, 5, 是], [13, 15, 4, 4, 是], [14, 0, 2, 2, 是], [14, 1, 1, 1, 是], [14, 2, 0, 0, 是], [14, 3, 15, 15, 是], [14, 4, 14, 14, 是], [14, 5, 13, 13, 是], [14, 6, 12, 12, 是], [14, 7, 11, 11, 是], [14, 8, 10, 10, 是], [14, 9, 9, 9, 是], [14, 10, 8, 8, 是], [14, 11, 7, 7, 是], [14, 12, 6, 6, 是], [14, 13, 5, 5, 是], [14, 14, 4, 4, 是], [14, 15, 3, 3, 是], [15, 0, 1, 1, 是], [15, 1, 0, 0, 是], [15, 2, 15, 15, 是], [15, 3, 14, 14, 是], [15, 4, 13, 13, 是], [15, 5, 12, 12, 是], [15, 6, 11, 11, 是], [15, 7, 10, 10, 是], [15, 8, 9, 9, 是], [15, 9, 8, 8, 是], [15, 10, 7, 7, 是], [15, 11, 6, 6, 是], [15, 12, 5, 5, 是], [15, 13, 4, 4, 是], [15, 14, 3, 3, 是], [15, 15, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(16, 16, [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 15, 15, 是], [0, 2, 14, 14, 是], [0, 3, 13, 13, 是], [0, 4, 12, 12, 是], [0, 5, 11, 11, 是], [0, 6, 10, 10, 是], [0, 7, 9, 9, 是], [0, 8, 8, 8, 是], [0, 9, 7, 7, 是], [0, 10, 6, 6, 是], [0, 11, 5, 5, 是], [0, 12, 4, 4, 是], [0, 13, 3, 3, 是], [0, 14, 2, 2, 是], [0, 15, 1, 1, 是], [1, 0, 15, 15, 是], [1, 1, 14, 14, 是], [1, 2, 13, 13, 是], [1, 3, 12, 12, 是], [1, 4, 11, 11, 是], [1, 5, 10, 10, 是], [1, 6, 9, 9, 是], [1, 7, 8, 8, 是], [1, 8, 7, 7, 是], [1, 9, 6, 6, 是], [1, 10, 5, 5, 是], [1, 11, 4, 4, 是], [1, 12, 3, 3, 是], [1, 13, 2, 2, 是], [1, 14, 1, 1, 是], [1, 15, 0, 0, 是], [2, 0, 14, 14, 是], [2, 1, 13, 13, 是], [2, 2, 12, 12, 是], [2, 3, 11, 11, 是], [2, 4, 10, 10, 是], [2, 5, 9, 9, 是], [2, 6, 8, 8, 是], [2, 7, 7, 7, 是], [2, 8, 6, 6, 是], [2, 9, 5, 5, 是], [2, 10, 4, 4, 是], [2, 11, 3, 3, 是], [2, 12, 2, 2, 是], [2, 13, 1, 1, 是], [2, 14, 0, 0, 是], [2, 15, 15, 15, 是], [3, 0, 13, 13, 是], [3, 1, 12, 12, 是], [3, 2, 11, 11, 是], [3, 3, 10, 10, 是], [3, 4, 9, 9, 是], [3, 5, 8, 8, 是], [3, 6, 7, 7, 是], [3, 7, 6, 6, 是], [3, 8, 5, 5, 是], [3, 9, 4, 4, 是], [3, 10, 3, 3, 是], [3, 11, 2, 2, 是], [3, 12, 1, 1, 是], [3, 13, 0, 0, 是], [3, 14, 15, 15, 是], [3, 15, 14, 14, 是], [4, 0, 12, 12, 是], [4, 1, 11, 11, 是], [4, 2, 10, 10, 是], [4, 3, 9, 9, 是], [4, 4, 8, 8, 是], [4, 5, 7, 7, 是], [4, 6, 6, 6, 是], [4, 7, 5, 5, 是], [4, 8, 4, 4, 是], [4, 9, 3, 3, 是], [4, 10, 2, 2, 是], [4, 11, 1, 1, 是], [4, 12, 0, 0, 是], [4, 13, 15, 15, 是], [4, 14, 14, 14, 是], [4, 15, 13, 13, 是], [5, 0, 11, 11, 是], [5, 1, 10, 10, 是], [5, 2, 9, 9, 是], [5, 3, 8, 8, 是], [5, 4, 7, 7, 是], [5, 5, 6, 6, 是], [5, 6, 5, 5, 是], [5, 7, 4, 4, 是], [5, 8, 3, 3, 是], [5, 9, 2, 2, 是], [5, 10, 1, 1, 是], [5, 11, 0, 0, 是], [5, 12, 15, 15, 是], [5, 13, 14, 14, 是], [5, 14, 13, 13, 是], [5, 15, 12, 12, 是], [6, 0, 10, 10, 是], [6, 1, 9, 9, 是], [6, 2, 8, 8, 是], [6, 3, 7, 7, 是], [6, 4, 6, 6, 是], [6, 5, 5, 5, 是], [6, 6, 4, 4, 是], [6, 7, 3, 3, 是], [6, 8, 2, 2, 是], [6, 9, 1, 1, 是], [6, 10, 0, 0, 是], [6, 11, 15, 15, 是], [6, 12, 14, 14, 是], [6, 13, 13, 13, 是], [6, 14, 12, 12, 是], [6, 15, 11, 11, 是], [7, 0, 9, 9, 是], [7, 1, 8, 8, 是], [7, 2, 7, 7, 是], [7, 3, 6, 6, 是], [7, 4, 5, 5, 是], [7, 5, 4, 4, 是], [7, 6, 3, 3, 是], [7, 7, 2, 2, 是], [7, 8, 1, 1, 是], [7, 9, 0, 0, 是], [7, 10, 15, 15, 是], [7, 11, 14, 14, 是], [7, 12, 13, 13, 是], [7, 13, 12, 12, 是], [7, 14, 11, 11, 是], [7, 15, 10, 10, 是], [8, 0, 8, 8, 是], [8, 1, 7, 7, 是], [8, 2, 6, 6, 是], [8, 3, 5, 5, 是], [8, 4, 4, 4, 是], [8, 5, 3, 3, 是], [8, 6, 2, 2, 是], [8, 7, 1, 1, 是], [8, 8, 0, 0, 是], [8, 9, 15, 15, 是], [8, 10, 14, 14, 是], [8, 11, 13, 13, 是], [8, 12, 12, 12, 是], [8, 13, 11, 11, 是], [8, 14, 10, 10, 是], [8, 15, 9, 9, 是], [9, 0, 7, 7, 是], [9, 1, 6, 6, 是], [9, 2, 5, 5, 是], [9, 3, 4, 4, 是], [9, 4, 3, 3, 是], [9, 5, 2, 2, 是], [9, 6, 1, 1, 是], [9, 7, 0, 0, 是], [9, 8, 15, 15, 是], [9, 9, 14, 14, 是], [9, 10, 13, 13, 是], [9, 11, 12, 12, 是], [9, 12, 11, 11, 是], [9, 13, 10, 10, 是], [9, 14, 9, 9, 是], [9, 15, 8, 8, 是], [10, 0, 6, 6, 是], [10, 1, 5, 5, 是], [10, 2, 4, 4, 是], [10, 3, 3, 3, 是], [10, 4, 2, 2, 是], [10, 5, 1, 1, 是], [10, 6, 0, 0, 是], [10, 7, 15, 15, 是], [10, 8, 14, 14, 是], [10, 9, 13, 13, 是], [10, 10, 12, 12, 是], [10, 11, 11, 11, 是], [10, 12, 10, 10, 是], [10, 13, 9, 9, 是], [10, 14, 8, 8, 是], [10, 15, 7, 7, 是], [11, 0, 5, 5, 是], [11, 1, 4, 4, 是], [11, 2, 3, 3, 是], [11, 3, 2, 2, 是], [11, 4, 1, 1, 是], [11, 5, 0, 0, 是], [11, 6, 15, 15, 是], [11, 7, 14, 14, 是], [11, 8, 13, 13, 是], [11, 9, 12, 12, 是], [11, 10, 11, 11, 是], [11, 11, 10, 10, 是], [11, 12, 9, 9, 是], [11, 13, 8, 8, 是], [11, 14, 7, 7, 是], [11, 15, 6, 6, 是], [12, 0, 4, 4, 是], [12, 1, 3, 3, 是], [12, 2, 2, 2, 是], [12, 3, 1, 1, 是], [12, 4, 0, 0, 是], [12, 5, 15, 15, 是], [12, 6, 14, 14, 是], [12, 7, 13, 13, 是], [12, 8, 12, 12, 是], [12, 9, 11, 11, 是], [12, 10, 10, 10, 是], [12, 11, 9, 9, 是], [12, 12, 8, 8, 是], [12, 13, 7, 7, 是], [12, 14, 6, 6, 是], [12, 15, 5, 5, 是], [13, 0, 3, 3, 是], [13, 1, 2, 2, 是], [13, 2, 1, 1, 是], [13, 3, 0, 0, 是], [13, 4, 15, 15, 是], [13, 5, 14, 14, 是], [13, 6, 13, 13, 是], [13, 7, 12, 12, 是], [13, 8, 11, 11, 是], [13, 9, 10, 10, 是], [13, 10, 9, 9, 是], [13, 11, 8, 8, 是], [13, 12, 7, 7, 是], [13, 13, 6, 6, 是], [13, 14, 5, 5, 是], [13, 15, 4, 4, 是], [14, 0, 2, 2, 是], [14, 1, 1, 1, 是], [14, 2, 0, 0, 是], [14, 3, 15, 15, 是], [14, 4, 14, 14, 是], [14, 5, 13, 13, 是], [14, 6, 12, 12, 是], [14, 7, 11, 11, 是], [14, 8, 10, 10, 是], [14, 9, 9, 9, 是], [14, 10, 8, 8, 是], [14, 11, 7, 7, 是], [14, 12, 6, 6, 是], [14, 13, 5, 5, 是], [14, 14, 4, 4, 是], [14, 15, 3, 3, 是], [15, 0, 1, 1, 是], [15, 1, 0, 0, 是], [15, 2, 15, 15, 是], [15, 3, 14, 14, 是], [15, 4, 13, 13, 是], [15, 5, 12, 12, 是], [15, 6, 11, 11, 是], [15, 7, 10, 10, 是], [15, 8, 9, 9, 是], [15, 9, 8, 8, 是], [15, 10, 7, 7, 是], [15, 11, 6, 6, 是], [15, 12, 5, 5, 是], [15, 13, 4, 4, 是], [15, 14, 3, 3, 是], [15, 15, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141014,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"target_identity":0,"mapping":[0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0015","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],\"target_identity\":0,\"mapping\":[0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 17, [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 16, 16, 是], [0, 2, 15, 15, 是], [0, 3, 14, 14, 是], [0, 4, 13, 13, 是], [0, 5, 12, 12, 是], [0, 6, 11, 11, 是], [0, 7, 10, 10, 是], [0, 8, 9, 9, 是], [0, 9, 8, 8, 是], [0, 10, 7, 7, 是], [0, 11, 6, 6, 是], [0, 12, 5, 5, 是], [0, 13, 4, 4, 是], [0, 14, 3, 3, 是], [0, 15, 2, 2, 是], [0, 16, 1, 1, 是], [1, 0, 16, 16, 是], [1, 1, 15, 15, 是], [1, 2, 14, 14, 是], [1, 3, 13, 13, 是], [1, 4, 12, 12, 是], [1, 5, 11, 11, 是], [1, 6, 10, 10, 是], [1, 7, 9, 9, 是], [1, 8, 8, 8, 是], [1, 9, 7, 7, 是], [1, 10, 6, 6, 是], [1, 11, 5, 5, 是], [1, 12, 4, 4, 是], [1, 13, 3, 3, 是], [1, 14, 2, 2, 是], [1, 15, 1, 1, 是], [1, 16, 0, 0, 是], [2, 0, 15, 15, 是], [2, 1, 14, 14, 是], [2, 2, 13, 13, 是], [2, 3, 12, 12, 是], [2, 4, 11, 11, 是], [2, 5, 10, 10, 是], [2, 6, 9, 9, 是], [2, 7, 8, 8, 是], [2, 8, 7, 7, 是], [2, 9, 6, 6, 是], [2, 10, 5, 5, 是], [2, 11, 4, 4, 是], [2, 12, 3, 3, 是], [2, 13, 2, 2, 是], [2, 14, 1, 1, 是], [2, 15, 0, 0, 是], [2, 16, 16, 16, 是], [3, 0, 14, 14, 是], [3, 1, 13, 13, 是], [3, 2, 12, 12, 是], [3, 3, 11, 11, 是], [3, 4, 10, 10, 是], [3, 5, 9, 9, 是], [3, 6, 8, 8, 是], [3, 7, 7, 7, 是], [3, 8, 6, 6, 是], [3, 9, 5, 5, 是], [3, 10, 4, 4, 是], [3, 11, 3, 3, 是], [3, 12, 2, 2, 是], [3, 13, 1, 1, 是], [3, 14, 0, 0, 是], [3, 15, 16, 16, 是], [3, 16, 15, 15, 是], [4, 0, 13, 13, 是], [4, 1, 12, 12, 是], [4, 2, 11, 11, 是], [4, 3, 10, 10, 是], [4, 4, 9, 9, 是], [4, 5, 8, 8, 是], [4, 6, 7, 7, 是], [4, 7, 6, 6, 是], [4, 8, 5, 5, 是], [4, 9, 4, 4, 是], [4, 10, 3, 3, 是], [4, 11, 2, 2, 是], [4, 12, 1, 1, 是], [4, 13, 0, 0, 是], [4, 14, 16, 16, 是], [4, 15, 15, 15, 是], [4, 16, 14, 14, 是], [5, 0, 12, 12, 是], [5, 1, 11, 11, 是], [5, 2, 10, 10, 是], [5, 3, 9, 9, 是], [5, 4, 8, 8, 是], [5, 5, 7, 7, 是], [5, 6, 6, 6, 是], [5, 7, 5, 5, 是], [5, 8, 4, 4, 是], [5, 9, 3, 3, 是], [5, 10, 2, 2, 是], [5, 11, 1, 1, 是], [5, 12, 0, 0, 是], [5, 13, 16, 16, 是], [5, 14, 15, 15, 是], [5, 15, 14, 14, 是], [5, 16, 13, 13, 是], [6, 0, 11, 11, 是], [6, 1, 10, 10, 是], [6, 2, 9, 9, 是], [6, 3, 8, 8, 是], [6, 4, 7, 7, 是], [6, 5, 6, 6, 是], [6, 6, 5, 5, 是], [6, 7, 4, 4, 是], [6, 8, 3, 3, 是], [6, 9, 2, 2, 是], [6, 10, 1, 1, 是], [6, 11, 0, 0, 是], [6, 12, 16, 16, 是], [6, 13, 15, 15, 是], [6, 14, 14, 14, 是], [6, 15, 13, 13, 是], [6, 16, 12, 12, 是], [7, 0, 10, 10, 是], [7, 1, 9, 9, 是], [7, 2, 8, 8, 是], [7, 3, 7, 7, 是], [7, 4, 6, 6, 是], [7, 5, 5, 5, 是], [7, 6, 4, 4, 是], [7, 7, 3, 3, 是], [7, 8, 2, 2, 是], [7, 9, 1, 1, 是], [7, 10, 0, 0, 是], [7, 11, 16, 16, 是], [7, 12, 15, 15, 是], [7, 13, 14, 14, 是], [7, 14, 13, 13, 是], [7, 15, 12, 12, 是], [7, 16, 11, 11, 是], [8, 0, 9, 9, 是], [8, 1, 8, 8, 是], [8, 2, 7, 7, 是], [8, 3, 6, 6, 是], [8, 4, 5, 5, 是], [8, 5, 4, 4, 是], [8, 6, 3, 3, 是], [8, 7, 2, 2, 是], [8, 8, 1, 1, 是], [8, 9, 0, 0, 是], [8, 10, 16, 16, 是], [8, 11, 15, 15, 是], [8, 12, 14, 14, 是], [8, 13, 13, 13, 是], [8, 14, 12, 12, 是], [8, 15, 11, 11, 是], [8, 16, 10, 10, 是], [9, 0, 8, 8, 是], [9, 1, 7, 7, 是], [9, 2, 6, 6, 是], [9, 3, 5, 5, 是], [9, 4, 4, 4, 是], [9, 5, 3, 3, 是], [9, 6, 2, 2, 是], [9, 7, 1, 1, 是], [9, 8, 0, 0, 是], [9, 9, 16, 16, 是], [9, 10, 15, 15, 是], [9, 11, 14, 14, 是], [9, 12, 13, 13, 是], [9, 13, 12, 12, 是], [9, 14, 11, 11, 是], [9, 15, 10, 10, 是], [9, 16, 9, 9, 是], [10, 0, 7, 7, 是], [10, 1, 6, 6, 是], [10, 2, 5, 5, 是], [10, 3, 4, 4, 是], [10, 4, 3, 3, 是], [10, 5, 2, 2, 是], [10, 6, 1, 1, 是], [10, 7, 0, 0, 是], [10, 8, 16, 16, 是], [10, 9, 15, 15, 是], [10, 10, 14, 14, 是], [10, 11, 13, 13, 是], [10, 12, 12, 12, 是], [10, 13, 11, 11, 是], [10, 14, 10, 10, 是], [10, 15, 9, 9, 是], [10, 16, 8, 8, 是], [11, 0, 6, 6, 是], [11, 1, 5, 5, 是], [11, 2, 4, 4, 是], [11, 3, 3, 3, 是], [11, 4, 2, 2, 是], [11, 5, 1, 1, 是], [11, 6, 0, 0, 是], [11, 7, 16, 16, 是], [11, 8, 15, 15, 是], [11, 9, 14, 14, 是], [11, 10, 13, 13, 是], [11, 11, 12, 12, 是], [11, 12, 11, 11, 是], [11, 13, 10, 10, 是], [11, 14, 9, 9, 是], [11, 15, 8, 8, 是], [11, 16, 7, 7, 是], [12, 0, 5, 5, 是], [12, 1, 4, 4, 是], [12, 2, 3, 3, 是], [12, 3, 2, 2, 是], [12, 4, 1, 1, 是], [12, 5, 0, 0, 是], [12, 6, 16, 16, 是], [12, 7, 15, 15, 是], [12, 8, 14, 14, 是], [12, 9, 13, 13, 是], [12, 10, 12, 12, 是], [12, 11, 11, 11, 是], [12, 12, 10, 10, 是], [12, 13, 9, 9, 是], [12, 14, 8, 8, 是], [12, 15, 7, 7, 是], [12, 16, 6, 6, 是], [13, 0, 4, 4, 是], [13, 1, 3, 3, 是], [13, 2, 2, 2, 是], [13, 3, 1, 1, 是], [13, 4, 0, 0, 是], [13, 5, 16, 16, 是], [13, 6, 15, 15, 是], [13, 7, 14, 14, 是], [13, 8, 13, 13, 是], [13, 9, 12, 12, 是], [13, 10, 11, 11, 是], [13, 11, 10, 10, 是], [13, 12, 9, 9, 是], [13, 13, 8, 8, 是], [13, 14, 7, 7, 是], [13, 15, 6, 6, 是], [13, 16, 5, 5, 是], [14, 0, 3, 3, 是], [14, 1, 2, 2, 是], [14, 2, 1, 1, 是], [14, 3, 0, 0, 是], [14, 4, 16, 16, 是], [14, 5, 15, 15, 是], [14, 6, 14, 14, 是], [14, 7, 13, 13, 是], [14, 8, 12, 12, 是], [14, 9, 11, 11, 是], [14, 10, 10, 10, 是], [14, 11, 9, 9, 是], [14, 12, 8, 8, 是], [14, 13, 7, 7, 是], [14, 14, 6, 6, 是], [14, 15, 5, 5, 是], [14, 16, 4, 4, 是], [15, 0, 2, 2, 是], [15, 1, 1, 1, 是], [15, 2, 0, 0, 是], [15, 3, 16, 16, 是], [15, 4, 15, 15, 是], [15, 5, 14, 14, 是], [15, 6, 13, 13, 是], [15, 7, 12, 12, 是], [15, 8, 11, 11, 是], [15, 9, 10, 10, 是], [15, 10, 9, 9, 是], [15, 11, 8, 8, 是], [15, 12, 7, 7, 是], [15, 13, 6, 6, 是], [15, 14, 5, 5, 是], [15, 15, 4, 4, 是], [15, 16, 3, 3, 是], [16, 0, 1, 1, 是], [16, 1, 0, 0, 是], [16, 2, 16, 16, 是], [16, 3, 15, 15, 是], [16, 4, 14, 14, 是], [16, 5, 13, 13, 是], [16, 6, 12, 12, 是], [16, 7, 11, 11, 是], [16, 8, 10, 10, 是], [16, 9, 9, 9, 是], [16, 10, 8, 8, 是], [16, 11, 7, 7, 是], [16, 12, 6, 6, 是], [16, 13, 5, 5, 是], [16, 14, 4, 4, 是], [16, 15, 3, 3, 是], [16, 16, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(17, 17, [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 16, 16, 是], [0, 2, 15, 15, 是], [0, 3, 14, 14, 是], [0, 4, 13, 13, 是], [0, 5, 12, 12, 是], [0, 6, 11, 11, 是], [0, 7, 10, 10, 是], [0, 8, 9, 9, 是], [0, 9, 8, 8, 是], [0, 10, 7, 7, 是], [0, 11, 6, 6, 是], [0, 12, 5, 5, 是], [0, 13, 4, 4, 是], [0, 14, 3, 3, 是], [0, 15, 2, 2, 是], [0, 16, 1, 1, 是], [1, 0, 16, 16, 是], [1, 1, 15, 15, 是], [1, 2, 14, 14, 是], [1, 3, 13, 13, 是], [1, 4, 12, 12, 是], [1, 5, 11, 11, 是], [1, 6, 10, 10, 是], [1, 7, 9, 9, 是], [1, 8, 8, 8, 是], [1, 9, 7, 7, 是], [1, 10, 6, 6, 是], [1, 11, 5, 5, 是], [1, 12, 4, 4, 是], [1, 13, 3, 3, 是], [1, 14, 2, 2, 是], [1, 15, 1, 1, 是], [1, 16, 0, 0, 是], [2, 0, 15, 15, 是], [2, 1, 14, 14, 是], [2, 2, 13, 13, 是], [2, 3, 12, 12, 是], [2, 4, 11, 11, 是], [2, 5, 10, 10, 是], [2, 6, 9, 9, 是], [2, 7, 8, 8, 是], [2, 8, 7, 7, 是], [2, 9, 6, 6, 是], [2, 10, 5, 5, 是], [2, 11, 4, 4, 是], [2, 12, 3, 3, 是], [2, 13, 2, 2, 是], [2, 14, 1, 1, 是], [2, 15, 0, 0, 是], [2, 16, 16, 16, 是], [3, 0, 14, 14, 是], [3, 1, 13, 13, 是], [3, 2, 12, 12, 是], [3, 3, 11, 11, 是], [3, 4, 10, 10, 是], [3, 5, 9, 9, 是], [3, 6, 8, 8, 是], [3, 7, 7, 7, 是], [3, 8, 6, 6, 是], [3, 9, 5, 5, 是], [3, 10, 4, 4, 是], [3, 11, 3, 3, 是], [3, 12, 2, 2, 是], [3, 13, 1, 1, 是], [3, 14, 0, 0, 是], [3, 15, 16, 16, 是], [3, 16, 15, 15, 是], [4, 0, 13, 13, 是], [4, 1, 12, 12, 是], [4, 2, 11, 11, 是], [4, 3, 10, 10, 是], [4, 4, 9, 9, 是], [4, 5, 8, 8, 是], [4, 6, 7, 7, 是], [4, 7, 6, 6, 是], [4, 8, 5, 5, 是], [4, 9, 4, 4, 是], [4, 10, 3, 3, 是], [4, 11, 2, 2, 是], [4, 12, 1, 1, 是], [4, 13, 0, 0, 是], [4, 14, 16, 16, 是], [4, 15, 15, 15, 是], [4, 16, 14, 14, 是], [5, 0, 12, 12, 是], [5, 1, 11, 11, 是], [5, 2, 10, 10, 是], [5, 3, 9, 9, 是], [5, 4, 8, 8, 是], [5, 5, 7, 7, 是], [5, 6, 6, 6, 是], [5, 7, 5, 5, 是], [5, 8, 4, 4, 是], [5, 9, 3, 3, 是], [5, 10, 2, 2, 是], [5, 11, 1, 1, 是], [5, 12, 0, 0, 是], [5, 13, 16, 16, 是], [5, 14, 15, 15, 是], [5, 15, 14, 14, 是], [5, 16, 13, 13, 是], [6, 0, 11, 11, 是], [6, 1, 10, 10, 是], [6, 2, 9, 9, 是], [6, 3, 8, 8, 是], [6, 4, 7, 7, 是], [6, 5, 6, 6, 是], [6, 6, 5, 5, 是], [6, 7, 4, 4, 是], [6, 8, 3, 3, 是], [6, 9, 2, 2, 是], [6, 10, 1, 1, 是], [6, 11, 0, 0, 是], [6, 12, 16, 16, 是], [6, 13, 15, 15, 是], [6, 14, 14, 14, 是], [6, 15, 13, 13, 是], [6, 16, 12, 12, 是], [7, 0, 10, 10, 是], [7, 1, 9, 9, 是], [7, 2, 8, 8, 是], [7, 3, 7, 7, 是], [7, 4, 6, 6, 是], [7, 5, 5, 5, 是], [7, 6, 4, 4, 是], [7, 7, 3, 3, 是], [7, 8, 2, 2, 是], [7, 9, 1, 1, 是], [7, 10, 0, 0, 是], [7, 11, 16, 16, 是], [7, 12, 15, 15, 是], [7, 13, 14, 14, 是], [7, 14, 13, 13, 是], [7, 15, 12, 12, 是], [7, 16, 11, 11, 是], [8, 0, 9, 9, 是], [8, 1, 8, 8, 是], [8, 2, 7, 7, 是], [8, 3, 6, 6, 是], [8, 4, 5, 5, 是], [8, 5, 4, 4, 是], [8, 6, 3, 3, 是], [8, 7, 2, 2, 是], [8, 8, 1, 1, 是], [8, 9, 0, 0, 是], [8, 10, 16, 16, 是], [8, 11, 15, 15, 是], [8, 12, 14, 14, 是], [8, 13, 13, 13, 是], [8, 14, 12, 12, 是], [8, 15, 11, 11, 是], [8, 16, 10, 10, 是], [9, 0, 8, 8, 是], [9, 1, 7, 7, 是], [9, 2, 6, 6, 是], [9, 3, 5, 5, 是], [9, 4, 4, 4, 是], [9, 5, 3, 3, 是], [9, 6, 2, 2, 是], [9, 7, 1, 1, 是], [9, 8, 0, 0, 是], [9, 9, 16, 16, 是], [9, 10, 15, 15, 是], [9, 11, 14, 14, 是], [9, 12, 13, 13, 是], [9, 13, 12, 12, 是], [9, 14, 11, 11, 是], [9, 15, 10, 10, 是], [9, 16, 9, 9, 是], [10, 0, 7, 7, 是], [10, 1, 6, 6, 是], [10, 2, 5, 5, 是], [10, 3, 4, 4, 是], [10, 4, 3, 3, 是], [10, 5, 2, 2, 是], [10, 6, 1, 1, 是], [10, 7, 0, 0, 是], [10, 8, 16, 16, 是], [10, 9, 15, 15, 是], [10, 10, 14, 14, 是], [10, 11, 13, 13, 是], [10, 12, 12, 12, 是], [10, 13, 11, 11, 是], [10, 14, 10, 10, 是], [10, 15, 9, 9, 是], [10, 16, 8, 8, 是], [11, 0, 6, 6, 是], [11, 1, 5, 5, 是], [11, 2, 4, 4, 是], [11, 3, 3, 3, 是], [11, 4, 2, 2, 是], [11, 5, 1, 1, 是], [11, 6, 0, 0, 是], [11, 7, 16, 16, 是], [11, 8, 15, 15, 是], [11, 9, 14, 14, 是], [11, 10, 13, 13, 是], [11, 11, 12, 12, 是], [11, 12, 11, 11, 是], [11, 13, 10, 10, 是], [11, 14, 9, 9, 是], [11, 15, 8, 8, 是], [11, 16, 7, 7, 是], [12, 0, 5, 5, 是], [12, 1, 4, 4, 是], [12, 2, 3, 3, 是], [12, 3, 2, 2, 是], [12, 4, 1, 1, 是], [12, 5, 0, 0, 是], [12, 6, 16, 16, 是], [12, 7, 15, 15, 是], [12, 8, 14, 14, 是], [12, 9, 13, 13, 是], [12, 10, 12, 12, 是], [12, 11, 11, 11, 是], [12, 12, 10, 10, 是], [12, 13, 9, 9, 是], [12, 14, 8, 8, 是], [12, 15, 7, 7, 是], [12, 16, 6, 6, 是], [13, 0, 4, 4, 是], [13, 1, 3, 3, 是], [13, 2, 2, 2, 是], [13, 3, 1, 1, 是], [13, 4, 0, 0, 是], [13, 5, 16, 16, 是], [13, 6, 15, 15, 是], [13, 7, 14, 14, 是], [13, 8, 13, 13, 是], [13, 9, 12, 12, 是], [13, 10, 11, 11, 是], [13, 11, 10, 10, 是], [13, 12, 9, 9, 是], [13, 13, 8, 8, 是], [13, 14, 7, 7, 是], [13, 15, 6, 6, 是], [13, 16, 5, 5, 是], [14, 0, 3, 3, 是], [14, 1, 2, 2, 是], [14, 2, 1, 1, 是], [14, 3, 0, 0, 是], [14, 4, 16, 16, 是], [14, 5, 15, 15, 是], [14, 6, 14, 14, 是], [14, 7, 13, 13, 是], [14, 8, 12, 12, 是], [14, 9, 11, 11, 是], [14, 10, 10, 10, 是], [14, 11, 9, 9, 是], [14, 12, 8, 8, 是], [14, 13, 7, 7, 是], [14, 14, 6, 6, 是], [14, 15, 5, 5, 是], [14, 16, 4, 4, 是], [15, 0, 2, 2, 是], [15, 1, 1, 1, 是], [15, 2, 0, 0, 是], [15, 3, 16, 16, 是], [15, 4, 15, 15, 是], [15, 5, 14, 14, 是], [15, 6, 13, 13, 是], [15, 7, 12, 12, 是], [15, 8, 11, 11, 是], [15, 9, 10, 10, 是], [15, 10, 9, 9, 是], [15, 11, 8, 8, 是], [15, 12, 7, 7, 是], [15, 13, 6, 6, 是], [15, 14, 5, 5, 是], [15, 15, 4, 4, 是], [15, 16, 3, 3, 是], [16, 0, 1, 1, 是], [16, 1, 0, 0, 是], [16, 2, 16, 16, 是], [16, 3, 15, 15, 是], [16, 4, 14, 14, 是], [16, 5, 13, 13, 是], [16, 6, 12, 12, 是], [16, 7, 11, 11, 是], [16, 8, 10, 10, 是], [16, 9, 9, 9, 是], [16, 10, 8, 8, 是], [16, 11, 7, 7, 是], [16, 12, 6, 6, 是], [16, 13, 5, 5, 是], [16, 14, 4, 4, 是], [16, 15, 3, 3, 是], [16, 16, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141015,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],"target_identity":0,"mapping":[0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0016","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"mapping\":[0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 18, [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 17, 17, 是], [0, 2, 16, 16, 是], [0, 3, 15, 15, 是], [0, 4, 14, 14, 是], [0, 5, 13, 13, 是], [0, 6, 12, 12, 是], [0, 7, 11, 11, 是], [0, 8, 10, 10, 是], [0, 9, 9, 9, 是], [0, 10, 8, 8, 是], [0, 11, 7, 7, 是], [0, 12, 6, 6, 是], [0, 13, 5, 5, 是], [0, 14, 4, 4, 是], [0, 15, 3, 3, 是], [0, 16, 2, 2, 是], [0, 17, 1, 1, 是], [1, 0, 17, 17, 是], [1, 1, 16, 16, 是], [1, 2, 15, 15, 是], [1, 3, 14, 14, 是], [1, 4, 13, 13, 是], [1, 5, 12, 12, 是], [1, 6, 11, 11, 是], [1, 7, 10, 10, 是], [1, 8, 9, 9, 是], [1, 9, 8, 8, 是], [1, 10, 7, 7, 是], [1, 11, 6, 6, 是], [1, 12, 5, 5, 是], [1, 13, 4, 4, 是], [1, 14, 3, 3, 是], [1, 15, 2, 2, 是], [1, 16, 1, 1, 是], [1, 17, 0, 0, 是], [2, 0, 16, 16, 是], [2, 1, 15, 15, 是], [2, 2, 14, 14, 是], [2, 3, 13, 13, 是], [2, 4, 12, 12, 是], [2, 5, 11, 11, 是], [2, 6, 10, 10, 是], [2, 7, 9, 9, 是], [2, 8, 8, 8, 是], [2, 9, 7, 7, 是], [2, 10, 6, 6, 是], [2, 11, 5, 5, 是], [2, 12, 4, 4, 是], [2, 13, 3, 3, 是], [2, 14, 2, 2, 是], [2, 15, 1, 1, 是], [2, 16, 0, 0, 是], [2, 17, 17, 17, 是], [3, 0, 15, 15, 是], [3, 1, 14, 14, 是], [3, 2, 13, 13, 是], [3, 3, 12, 12, 是], [3, 4, 11, 11, 是], [3, 5, 10, 10, 是], [3, 6, 9, 9, 是], [3, 7, 8, 8, 是], [3, 8, 7, 7, 是], [3, 9, 6, 6, 是], [3, 10, 5, 5, 是], [3, 11, 4, 4, 是], [3, 12, 3, 3, 是], [3, 13, 2, 2, 是], [3, 14, 1, 1, 是], [3, 15, 0, 0, 是], [3, 16, 17, 17, 是], [3, 17, 16, 16, 是], [4, 0, 14, 14, 是], [4, 1, 13, 13, 是], [4, 2, 12, 12, 是], [4, 3, 11, 11, 是], [4, 4, 10, 10, 是], [4, 5, 9, 9, 是], [4, 6, 8, 8, 是], [4, 7, 7, 7, 是], [4, 8, 6, 6, 是], [4, 9, 5, 5, 是], [4, 10, 4, 4, 是], [4, 11, 3, 3, 是], [4, 12, 2, 2, 是], [4, 13, 1, 1, 是], [4, 14, 0, 0, 是], [4, 15, 17, 17, 是], [4, 16, 16, 16, 是], [4, 17, 15, 15, 是], [5, 0, 13, 13, 是], [5, 1, 12, 12, 是], [5, 2, 11, 11, 是], [5, 3, 10, 10, 是], [5, 4, 9, 9, 是], [5, 5, 8, 8, 是], [5, 6, 7, 7, 是], [5, 7, 6, 6, 是], [5, 8, 5, 5, 是], [5, 9, 4, 4, 是], [5, 10, 3, 3, 是], [5, 11, 2, 2, 是], [5, 12, 1, 1, 是], [5, 13, 0, 0, 是], [5, 14, 17, 17, 是], [5, 15, 16, 16, 是], [5, 16, 15, 15, 是], [5, 17, 14, 14, 是], [6, 0, 12, 12, 是], [6, 1, 11, 11, 是], [6, 2, 10, 10, 是], [6, 3, 9, 9, 是], [6, 4, 8, 8, 是], [6, 5, 7, 7, 是], [6, 6, 6, 6, 是], [6, 7, 5, 5, 是], [6, 8, 4, 4, 是], [6, 9, 3, 3, 是], [6, 10, 2, 2, 是], [6, 11, 1, 1, 是], [6, 12, 0, 0, 是], [6, 13, 17, 17, 是], [6, 14, 16, 16, 是], [6, 15, 15, 15, 是], [6, 16, 14, 14, 是], [6, 17, 13, 13, 是], [7, 0, 11, 11, 是], [7, 1, 10, 10, 是], [7, 2, 9, 9, 是], [7, 3, 8, 8, 是], [7, 4, 7, 7, 是], [7, 5, 6, 6, 是], [7, 6, 5, 5, 是], [7, 7, 4, 4, 是], [7, 8, 3, 3, 是], [7, 9, 2, 2, 是], [7, 10, 1, 1, 是], [7, 11, 0, 0, 是], [7, 12, 17, 17, 是], [7, 13, 16, 16, 是], [7, 14, 15, 15, 是], [7, 15, 14, 14, 是], [7, 16, 13, 13, 是], [7, 17, 12, 12, 是], [8, 0, 10, 10, 是], [8, 1, 9, 9, 是], [8, 2, 8, 8, 是], [8, 3, 7, 7, 是], [8, 4, 6, 6, 是], [8, 5, 5, 5, 是], [8, 6, 4, 4, 是], [8, 7, 3, 3, 是], [8, 8, 2, 2, 是], [8, 9, 1, 1, 是], [8, 10, 0, 0, 是], [8, 11, 17, 17, 是], [8, 12, 16, 16, 是], [8, 13, 15, 15, 是], [8, 14, 14, 14, 是], [8, 15, 13, 13, 是], [8, 16, 12, 12, 是], [8, 17, 11, 11, 是], [9, 0, 9, 9, 是], [9, 1, 8, 8, 是], [9, 2, 7, 7, 是], [9, 3, 6, 6, 是], [9, 4, 5, 5, 是], [9, 5, 4, 4, 是], [9, 6, 3, 3, 是], [9, 7, 2, 2, 是], [9, 8, 1, 1, 是], [9, 9, 0, 0, 是], [9, 10, 17, 17, 是], [9, 11, 16, 16, 是], [9, 12, 15, 15, 是], [9, 13, 14, 14, 是], [9, 14, 13, 13, 是], [9, 15, 12, 12, 是], [9, 16, 11, 11, 是], [9, 17, 10, 10, 是], [10, 0, 8, 8, 是], [10, 1, 7, 7, 是], [10, 2, 6, 6, 是], [10, 3, 5, 5, 是], [10, 4, 4, 4, 是], [10, 5, 3, 3, 是], [10, 6, 2, 2, 是], [10, 7, 1, 1, 是], [10, 8, 0, 0, 是], [10, 9, 17, 17, 是], [10, 10, 16, 16, 是], [10, 11, 15, 15, 是], [10, 12, 14, 14, 是], [10, 13, 13, 13, 是], [10, 14, 12, 12, 是], [10, 15, 11, 11, 是], [10, 16, 10, 10, 是], [10, 17, 9, 9, 是], [11, 0, 7, 7, 是], [11, 1, 6, 6, 是], [11, 2, 5, 5, 是], [11, 3, 4, 4, 是], [11, 4, 3, 3, 是], [11, 5, 2, 2, 是], [11, 6, 1, 1, 是], [11, 7, 0, 0, 是], [11, 8, 17, 17, 是], [11, 9, 16, 16, 是], [11, 10, 15, 15, 是], [11, 11, 14, 14, 是], [11, 12, 13, 13, 是], [11, 13, 12, 12, 是], [11, 14, 11, 11, 是], [11, 15, 10, 10, 是], [11, 16, 9, 9, 是], [11, 17, 8, 8, 是], [12, 0, 6, 6, 是], [12, 1, 5, 5, 是], [12, 2, 4, 4, 是], [12, 3, 3, 3, 是], [12, 4, 2, 2, 是], [12, 5, 1, 1, 是], [12, 6, 0, 0, 是], [12, 7, 17, 17, 是], [12, 8, 16, 16, 是], [12, 9, 15, 15, 是], [12, 10, 14, 14, 是], [12, 11, 13, 13, 是], [12, 12, 12, 12, 是], [12, 13, 11, 11, 是], [12, 14, 10, 10, 是], [12, 15, 9, 9, 是], [12, 16, 8, 8, 是], [12, 17, 7, 7, 是], [13, 0, 5, 5, 是], [13, 1, 4, 4, 是], [13, 2, 3, 3, 是], [13, 3, 2, 2, 是], [13, 4, 1, 1, 是], [13, 5, 0, 0, 是], [13, 6, 17, 17, 是], [13, 7, 16, 16, 是], [13, 8, 15, 15, 是], [13, 9, 14, 14, 是], [13, 10, 13, 13, 是], [13, 11, 12, 12, 是], [13, 12, 11, 11, 是], [13, 13, 10, 10, 是], [13, 14, 9, 9, 是], [13, 15, 8, 8, 是], [13, 16, 7, 7, 是], [13, 17, 6, 6, 是], [14, 0, 4, 4, 是], [14, 1, 3, 3, 是], [14, 2, 2, 2, 是], [14, 3, 1, 1, 是], [14, 4, 0, 0, 是], [14, 5, 17, 17, 是], [14, 6, 16, 16, 是], [14, 7, 15, 15, 是], [14, 8, 14, 14, 是], [14, 9, 13, 13, 是], [14, 10, 12, 12, 是], [14, 11, 11, 11, 是], [14, 12, 10, 10, 是], [14, 13, 9, 9, 是], [14, 14, 8, 8, 是], [14, 15, 7, 7, 是], [14, 16, 6, 6, 是], [14, 17, 5, 5, 是], [15, 0, 3, 3, 是], [15, 1, 2, 2, 是], [15, 2, 1, 1, 是], [15, 3, 0, 0, 是], [15, 4, 17, 17, 是], [15, 5, 16, 16, 是], [15, 6, 15, 15, 是], [15, 7, 14, 14, 是], [15, 8, 13, 13, 是], [15, 9, 12, 12, 是], [15, 10, 11, 11, 是], [15, 11, 10, 10, 是], [15, 12, 9, 9, 是], [15, 13, 8, 8, 是], [15, 14, 7, 7, 是], [15, 15, 6, 6, 是], [15, 16, 5, 5, 是], [15, 17, 4, 4, 是], [16, 0, 2, 2, 是], [16, 1, 1, 1, 是], [16, 2, 0, 0, 是], [16, 3, 17, 17, 是], [16, 4, 16, 16, 是], [16, 5, 15, 15, 是], [16, 6, 14, 14, 是], [16, 7, 13, 13, 是], [16, 8, 12, 12, 是], [16, 9, 11, 11, 是], [16, 10, 10, 10, 是], [16, 11, 9, 9, 是], [16, 12, 8, 8, 是], [16, 13, 7, 7, 是], [16, 14, 6, 6, 是], [16, 15, 5, 5, 是], [16, 16, 4, 4, 是], [16, 17, 3, 3, 是], [17, 0, 1, 1, 是], [17, 1, 0, 0, 是], [17, 2, 17, 17, 是], [17, 3, 16, 16, 是], [17, 4, 15, 15, 是], [17, 5, 14, 14, 是], [17, 6, 13, 13, 是], [17, 7, 12, 12, 是], [17, 8, 11, 11, 是], [17, 9, 10, 10, 是], [17, 10, 9, 9, 是], [17, 11, 8, 8, 是], [17, 12, 7, 7, 是], [17, 13, 6, 6, 是], [17, 14, 5, 5, 是], [17, 15, 4, 4, 是], [17, 16, 3, 3, 是], [17, 17, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(18, 18, [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 17, 17, 是], [0, 2, 16, 16, 是], [0, 3, 15, 15, 是], [0, 4, 14, 14, 是], [0, 5, 13, 13, 是], [0, 6, 12, 12, 是], [0, 7, 11, 11, 是], [0, 8, 10, 10, 是], [0, 9, 9, 9, 是], [0, 10, 8, 8, 是], [0, 11, 7, 7, 是], [0, 12, 6, 6, 是], [0, 13, 5, 5, 是], [0, 14, 4, 4, 是], [0, 15, 3, 3, 是], [0, 16, 2, 2, 是], [0, 17, 1, 1, 是], [1, 0, 17, 17, 是], [1, 1, 16, 16, 是], [1, 2, 15, 15, 是], [1, 3, 14, 14, 是], [1, 4, 13, 13, 是], [1, 5, 12, 12, 是], [1, 6, 11, 11, 是], [1, 7, 10, 10, 是], [1, 8, 9, 9, 是], [1, 9, 8, 8, 是], [1, 10, 7, 7, 是], [1, 11, 6, 6, 是], [1, 12, 5, 5, 是], [1, 13, 4, 4, 是], [1, 14, 3, 3, 是], [1, 15, 2, 2, 是], [1, 16, 1, 1, 是], [1, 17, 0, 0, 是], [2, 0, 16, 16, 是], [2, 1, 15, 15, 是], [2, 2, 14, 14, 是], [2, 3, 13, 13, 是], [2, 4, 12, 12, 是], [2, 5, 11, 11, 是], [2, 6, 10, 10, 是], [2, 7, 9, 9, 是], [2, 8, 8, 8, 是], [2, 9, 7, 7, 是], [2, 10, 6, 6, 是], [2, 11, 5, 5, 是], [2, 12, 4, 4, 是], [2, 13, 3, 3, 是], [2, 14, 2, 2, 是], [2, 15, 1, 1, 是], [2, 16, 0, 0, 是], [2, 17, 17, 17, 是], [3, 0, 15, 15, 是], [3, 1, 14, 14, 是], [3, 2, 13, 13, 是], [3, 3, 12, 12, 是], [3, 4, 11, 11, 是], [3, 5, 10, 10, 是], [3, 6, 9, 9, 是], [3, 7, 8, 8, 是], [3, 8, 7, 7, 是], [3, 9, 6, 6, 是], [3, 10, 5, 5, 是], [3, 11, 4, 4, 是], [3, 12, 3, 3, 是], [3, 13, 2, 2, 是], [3, 14, 1, 1, 是], [3, 15, 0, 0, 是], [3, 16, 17, 17, 是], [3, 17, 16, 16, 是], [4, 0, 14, 14, 是], [4, 1, 13, 13, 是], [4, 2, 12, 12, 是], [4, 3, 11, 11, 是], [4, 4, 10, 10, 是], [4, 5, 9, 9, 是], [4, 6, 8, 8, 是], [4, 7, 7, 7, 是], [4, 8, 6, 6, 是], [4, 9, 5, 5, 是], [4, 10, 4, 4, 是], [4, 11, 3, 3, 是], [4, 12, 2, 2, 是], [4, 13, 1, 1, 是], [4, 14, 0, 0, 是], [4, 15, 17, 17, 是], [4, 16, 16, 16, 是], [4, 17, 15, 15, 是], [5, 0, 13, 13, 是], [5, 1, 12, 12, 是], [5, 2, 11, 11, 是], [5, 3, 10, 10, 是], [5, 4, 9, 9, 是], [5, 5, 8, 8, 是], [5, 6, 7, 7, 是], [5, 7, 6, 6, 是], [5, 8, 5, 5, 是], [5, 9, 4, 4, 是], [5, 10, 3, 3, 是], [5, 11, 2, 2, 是], [5, 12, 1, 1, 是], [5, 13, 0, 0, 是], [5, 14, 17, 17, 是], [5, 15, 16, 16, 是], [5, 16, 15, 15, 是], [5, 17, 14, 14, 是], [6, 0, 12, 12, 是], [6, 1, 11, 11, 是], [6, 2, 10, 10, 是], [6, 3, 9, 9, 是], [6, 4, 8, 8, 是], [6, 5, 7, 7, 是], [6, 6, 6, 6, 是], [6, 7, 5, 5, 是], [6, 8, 4, 4, 是], [6, 9, 3, 3, 是], [6, 10, 2, 2, 是], [6, 11, 1, 1, 是], [6, 12, 0, 0, 是], [6, 13, 17, 17, 是], [6, 14, 16, 16, 是], [6, 15, 15, 15, 是], [6, 16, 14, 14, 是], [6, 17, 13, 13, 是], [7, 0, 11, 11, 是], [7, 1, 10, 10, 是], [7, 2, 9, 9, 是], [7, 3, 8, 8, 是], [7, 4, 7, 7, 是], [7, 5, 6, 6, 是], [7, 6, 5, 5, 是], [7, 7, 4, 4, 是], [7, 8, 3, 3, 是], [7, 9, 2, 2, 是], [7, 10, 1, 1, 是], [7, 11, 0, 0, 是], [7, 12, 17, 17, 是], [7, 13, 16, 16, 是], [7, 14, 15, 15, 是], [7, 15, 14, 14, 是], [7, 16, 13, 13, 是], [7, 17, 12, 12, 是], [8, 0, 10, 10, 是], [8, 1, 9, 9, 是], [8, 2, 8, 8, 是], [8, 3, 7, 7, 是], [8, 4, 6, 6, 是], [8, 5, 5, 5, 是], [8, 6, 4, 4, 是], [8, 7, 3, 3, 是], [8, 8, 2, 2, 是], [8, 9, 1, 1, 是], [8, 10, 0, 0, 是], [8, 11, 17, 17, 是], [8, 12, 16, 16, 是], [8, 13, 15, 15, 是], [8, 14, 14, 14, 是], [8, 15, 13, 13, 是], [8, 16, 12, 12, 是], [8, 17, 11, 11, 是], [9, 0, 9, 9, 是], [9, 1, 8, 8, 是], [9, 2, 7, 7, 是], [9, 3, 6, 6, 是], [9, 4, 5, 5, 是], [9, 5, 4, 4, 是], [9, 6, 3, 3, 是], [9, 7, 2, 2, 是], [9, 8, 1, 1, 是], [9, 9, 0, 0, 是], [9, 10, 17, 17, 是], [9, 11, 16, 16, 是], [9, 12, 15, 15, 是], [9, 13, 14, 14, 是], [9, 14, 13, 13, 是], [9, 15, 12, 12, 是], [9, 16, 11, 11, 是], [9, 17, 10, 10, 是], [10, 0, 8, 8, 是], [10, 1, 7, 7, 是], [10, 2, 6, 6, 是], [10, 3, 5, 5, 是], [10, 4, 4, 4, 是], [10, 5, 3, 3, 是], [10, 6, 2, 2, 是], [10, 7, 1, 1, 是], [10, 8, 0, 0, 是], [10, 9, 17, 17, 是], [10, 10, 16, 16, 是], [10, 11, 15, 15, 是], [10, 12, 14, 14, 是], [10, 13, 13, 13, 是], [10, 14, 12, 12, 是], [10, 15, 11, 11, 是], [10, 16, 10, 10, 是], [10, 17, 9, 9, 是], [11, 0, 7, 7, 是], [11, 1, 6, 6, 是], [11, 2, 5, 5, 是], [11, 3, 4, 4, 是], [11, 4, 3, 3, 是], [11, 5, 2, 2, 是], [11, 6, 1, 1, 是], [11, 7, 0, 0, 是], [11, 8, 17, 17, 是], [11, 9, 16, 16, 是], [11, 10, 15, 15, 是], [11, 11, 14, 14, 是], [11, 12, 13, 13, 是], [11, 13, 12, 12, 是], [11, 14, 11, 11, 是], [11, 15, 10, 10, 是], [11, 16, 9, 9, 是], [11, 17, 8, 8, 是], [12, 0, 6, 6, 是], [12, 1, 5, 5, 是], [12, 2, 4, 4, 是], [12, 3, 3, 3, 是], [12, 4, 2, 2, 是], [12, 5, 1, 1, 是], [12, 6, 0, 0, 是], [12, 7, 17, 17, 是], [12, 8, 16, 16, 是], [12, 9, 15, 15, 是], [12, 10, 14, 14, 是], [12, 11, 13, 13, 是], [12, 12, 12, 12, 是], [12, 13, 11, 11, 是], [12, 14, 10, 10, 是], [12, 15, 9, 9, 是], [12, 16, 8, 8, 是], [12, 17, 7, 7, 是], [13, 0, 5, 5, 是], [13, 1, 4, 4, 是], [13, 2, 3, 3, 是], [13, 3, 2, 2, 是], [13, 4, 1, 1, 是], [13, 5, 0, 0, 是], [13, 6, 17, 17, 是], [13, 7, 16, 16, 是], [13, 8, 15, 15, 是], [13, 9, 14, 14, 是], [13, 10, 13, 13, 是], [13, 11, 12, 12, 是], [13, 12, 11, 11, 是], [13, 13, 10, 10, 是], [13, 14, 9, 9, 是], [13, 15, 8, 8, 是], [13, 16, 7, 7, 是], [13, 17, 6, 6, 是], [14, 0, 4, 4, 是], [14, 1, 3, 3, 是], [14, 2, 2, 2, 是], [14, 3, 1, 1, 是], [14, 4, 0, 0, 是], [14, 5, 17, 17, 是], [14, 6, 16, 16, 是], [14, 7, 15, 15, 是], [14, 8, 14, 14, 是], [14, 9, 13, 13, 是], [14, 10, 12, 12, 是], [14, 11, 11, 11, 是], [14, 12, 10, 10, 是], [14, 13, 9, 9, 是], [14, 14, 8, 8, 是], [14, 15, 7, 7, 是], [14, 16, 6, 6, 是], [14, 17, 5, 5, 是], [15, 0, 3, 3, 是], [15, 1, 2, 2, 是], [15, 2, 1, 1, 是], [15, 3, 0, 0, 是], [15, 4, 17, 17, 是], [15, 5, 16, 16, 是], [15, 6, 15, 15, 是], [15, 7, 14, 14, 是], [15, 8, 13, 13, 是], [15, 9, 12, 12, 是], [15, 10, 11, 11, 是], [15, 11, 10, 10, 是], [15, 12, 9, 9, 是], [15, 13, 8, 8, 是], [15, 14, 7, 7, 是], [15, 15, 6, 6, 是], [15, 16, 5, 5, 是], [15, 17, 4, 4, 是], [16, 0, 2, 2, 是], [16, 1, 1, 1, 是], [16, 2, 0, 0, 是], [16, 3, 17, 17, 是], [16, 4, 16, 16, 是], [16, 5, 15, 15, 是], [16, 6, 14, 14, 是], [16, 7, 13, 13, 是], [16, 8, 12, 12, 是], [16, 9, 11, 11, 是], [16, 10, 10, 10, 是], [16, 11, 9, 9, 是], [16, 12, 8, 8, 是], [16, 13, 7, 7, 是], [16, 14, 6, 6, 是], [16, 15, 5, 5, 是], [16, 16, 4, 4, 是], [16, 17, 3, 3, 是], [17, 0, 1, 1, 是], [17, 1, 0, 0, 是], [17, 2, 17, 17, 是], [17, 3, 16, 16, 是], [17, 4, 15, 15, 是], [17, 5, 14, 14, 是], [17, 6, 13, 13, 是], [17, 7, 12, 12, 是], [17, 8, 11, 11, 是], [17, 9, 10, 10, 是], [17, 10, 9, 9, 是], [17, 11, 8, 8, 是], [17, 12, 7, 7, 是], [17, 13, 6, 6, 是], [17, 14, 5, 5, 是], [17, 15, 4, 4, 是], [17, 16, 3, 3, 是], [17, 17, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141016,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],"target_identity":0,"mapping":[0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0017","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],\"target_identity\":0,\"mapping\":[0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 19, [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 18, 18, 是], [0, 2, 17, 17, 是], [0, 3, 16, 16, 是], [0, 4, 15, 15, 是], [0, 5, 14, 14, 是], [0, 6, 13, 13, 是], [0, 7, 12, 12, 是], [0, 8, 11, 11, 是], [0, 9, 10, 10, 是], [0, 10, 9, 9, 是], [0, 11, 8, 8, 是], [0, 12, 7, 7, 是], [0, 13, 6, 6, 是], [0, 14, 5, 5, 是], [0, 15, 4, 4, 是], [0, 16, 3, 3, 是], [0, 17, 2, 2, 是], [0, 18, 1, 1, 是], [1, 0, 18, 18, 是], [1, 1, 17, 17, 是], [1, 2, 16, 16, 是], [1, 3, 15, 15, 是], [1, 4, 14, 14, 是], [1, 5, 13, 13, 是], [1, 6, 12, 12, 是], [1, 7, 11, 11, 是], [1, 8, 10, 10, 是], [1, 9, 9, 9, 是], [1, 10, 8, 8, 是], [1, 11, 7, 7, 是], [1, 12, 6, 6, 是], [1, 13, 5, 5, 是], [1, 14, 4, 4, 是], [1, 15, 3, 3, 是], [1, 16, 2, 2, 是], [1, 17, 1, 1, 是], [1, 18, 0, 0, 是], [2, 0, 17, 17, 是], [2, 1, 16, 16, 是], [2, 2, 15, 15, 是], [2, 3, 14, 14, 是], [2, 4, 13, 13, 是], [2, 5, 12, 12, 是], [2, 6, 11, 11, 是], [2, 7, 10, 10, 是], [2, 8, 9, 9, 是], [2, 9, 8, 8, 是], [2, 10, 7, 7, 是], [2, 11, 6, 6, 是], [2, 12, 5, 5, 是], [2, 13, 4, 4, 是], [2, 14, 3, 3, 是], [2, 15, 2, 2, 是], [2, 16, 1, 1, 是], [2, 17, 0, 0, 是], [2, 18, 18, 18, 是], [3, 0, 16, 16, 是], [3, 1, 15, 15, 是], [3, 2, 14, 14, 是], [3, 3, 13, 13, 是], [3, 4, 12, 12, 是], [3, 5, 11, 11, 是], [3, 6, 10, 10, 是], [3, 7, 9, 9, 是], [3, 8, 8, 8, 是], [3, 9, 7, 7, 是], [3, 10, 6, 6, 是], [3, 11, 5, 5, 是], [3, 12, 4, 4, 是], [3, 13, 3, 3, 是], [3, 14, 2, 2, 是], [3, 15, 1, 1, 是], [3, 16, 0, 0, 是], [3, 17, 18, 18, 是], [3, 18, 17, 17, 是], [4, 0, 15, 15, 是], [4, 1, 14, 14, 是], [4, 2, 13, 13, 是], [4, 3, 12, 12, 是], [4, 4, 11, 11, 是], [4, 5, 10, 10, 是], [4, 6, 9, 9, 是], [4, 7, 8, 8, 是], [4, 8, 7, 7, 是], [4, 9, 6, 6, 是], [4, 10, 5, 5, 是], [4, 11, 4, 4, 是], [4, 12, 3, 3, 是], [4, 13, 2, 2, 是], [4, 14, 1, 1, 是], [4, 15, 0, 0, 是], [4, 16, 18, 18, 是], [4, 17, 17, 17, 是], [4, 18, 16, 16, 是], [5, 0, 14, 14, 是], [5, 1, 13, 13, 是], [5, 2, 12, 12, 是], [5, 3, 11, 11, 是], [5, 4, 10, 10, 是], [5, 5, 9, 9, 是], [5, 6, 8, 8, 是], [5, 7, 7, 7, 是], [5, 8, 6, 6, 是], [5, 9, 5, 5, 是], [5, 10, 4, 4, 是], [5, 11, 3, 3, 是], [5, 12, 2, 2, 是], [5, 13, 1, 1, 是], [5, 14, 0, 0, 是], [5, 15, 18, 18, 是], [5, 16, 17, 17, 是], [5, 17, 16, 16, 是], [5, 18, 15, 15, 是], [6, 0, 13, 13, 是], [6, 1, 12, 12, 是], [6, 2, 11, 11, 是], [6, 3, 10, 10, 是], [6, 4, 9, 9, 是], [6, 5, 8, 8, 是], [6, 6, 7, 7, 是], [6, 7, 6, 6, 是], [6, 8, 5, 5, 是], [6, 9, 4, 4, 是], [6, 10, 3, 3, 是], [6, 11, 2, 2, 是], [6, 12, 1, 1, 是], [6, 13, 0, 0, 是], [6, 14, 18, 18, 是], [6, 15, 17, 17, 是], [6, 16, 16, 16, 是], [6, 17, 15, 15, 是], [6, 18, 14, 14, 是], [7, 0, 12, 12, 是], [7, 1, 11, 11, 是], [7, 2, 10, 10, 是], [7, 3, 9, 9, 是], [7, 4, 8, 8, 是], [7, 5, 7, 7, 是], [7, 6, 6, 6, 是], [7, 7, 5, 5, 是], [7, 8, 4, 4, 是], [7, 9, 3, 3, 是], [7, 10, 2, 2, 是], [7, 11, 1, 1, 是], [7, 12, 0, 0, 是], [7, 13, 18, 18, 是], [7, 14, 17, 17, 是], [7, 15, 16, 16, 是], [7, 16, 15, 15, 是], [7, 17, 14, 14, 是], [7, 18, 13, 13, 是], [8, 0, 11, 11, 是], [8, 1, 10, 10, 是], [8, 2, 9, 9, 是], [8, 3, 8, 8, 是], [8, 4, 7, 7, 是], [8, 5, 6, 6, 是], [8, 6, 5, 5, 是], [8, 7, 4, 4, 是], [8, 8, 3, 3, 是], [8, 9, 2, 2, 是], [8, 10, 1, 1, 是], [8, 11, 0, 0, 是], [8, 12, 18, 18, 是], [8, 13, 17, 17, 是], [8, 14, 16, 16, 是], [8, 15, 15, 15, 是], [8, 16, 14, 14, 是], [8, 17, 13, 13, 是], [8, 18, 12, 12, 是], [9, 0, 10, 10, 是], [9, 1, 9, 9, 是], [9, 2, 8, 8, 是], [9, 3, 7, 7, 是], [9, 4, 6, 6, 是], [9, 5, 5, 5, 是], [9, 6, 4, 4, 是], [9, 7, 3, 3, 是], [9, 8, 2, 2, 是], [9, 9, 1, 1, 是], [9, 10, 0, 0, 是], [9, 11, 18, 18, 是], [9, 12, 17, 17, 是], [9, 13, 16, 16, 是], [9, 14, 15, 15, 是], [9, 15, 14, 14, 是], [9, 16, 13, 13, 是], [9, 17, 12, 12, 是], [9, 18, 11, 11, 是], [10, 0, 9, 9, 是], [10, 1, 8, 8, 是], [10, 2, 7, 7, 是], [10, 3, 6, 6, 是], [10, 4, 5, 5, 是], [10, 5, 4, 4, 是], [10, 6, 3, 3, 是], [10, 7, 2, 2, 是], [10, 8, 1, 1, 是], [10, 9, 0, 0, 是], [10, 10, 18, 18, 是], [10, 11, 17, 17, 是], [10, 12, 16, 16, 是], [10, 13, 15, 15, 是], [10, 14, 14, 14, 是], [10, 15, 13, 13, 是], [10, 16, 12, 12, 是], [10, 17, 11, 11, 是], [10, 18, 10, 10, 是], [11, 0, 8, 8, 是], [11, 1, 7, 7, 是], [11, 2, 6, 6, 是], [11, 3, 5, 5, 是], [11, 4, 4, 4, 是], [11, 5, 3, 3, 是], [11, 6, 2, 2, 是], [11, 7, 1, 1, 是], [11, 8, 0, 0, 是], [11, 9, 18, 18, 是], [11, 10, 17, 17, 是], [11, 11, 16, 16, 是], [11, 12, 15, 15, 是], [11, 13, 14, 14, 是], [11, 14, 13, 13, 是], [11, 15, 12, 12, 是], [11, 16, 11, 11, 是], [11, 17, 10, 10, 是], [11, 18, 9, 9, 是], [12, 0, 7, 7, 是], [12, 1, 6, 6, 是], [12, 2, 5, 5, 是], [12, 3, 4, 4, 是], [12, 4, 3, 3, 是], [12, 5, 2, 2, 是], [12, 6, 1, 1, 是], [12, 7, 0, 0, 是], [12, 8, 18, 18, 是], [12, 9, 17, 17, 是], [12, 10, 16, 16, 是], [12, 11, 15, 15, 是], [12, 12, 14, 14, 是], [12, 13, 13, 13, 是], [12, 14, 12, 12, 是], [12, 15, 11, 11, 是], [12, 16, 10, 10, 是], [12, 17, 9, 9, 是], [12, 18, 8, 8, 是], [13, 0, 6, 6, 是], [13, 1, 5, 5, 是], [13, 2, 4, 4, 是], [13, 3, 3, 3, 是], [13, 4, 2, 2, 是], [13, 5, 1, 1, 是], [13, 6, 0, 0, 是], [13, 7, 18, 18, 是], [13, 8, 17, 17, 是], [13, 9, 16, 16, 是], [13, 10, 15, 15, 是], [13, 11, 14, 14, 是], [13, 12, 13, 13, 是], [13, 13, 12, 12, 是], [13, 14, 11, 11, 是], [13, 15, 10, 10, 是], [13, 16, 9, 9, 是], [13, 17, 8, 8, 是], [13, 18, 7, 7, 是], [14, 0, 5, 5, 是], [14, 1, 4, 4, 是], [14, 2, 3, 3, 是], [14, 3, 2, 2, 是], [14, 4, 1, 1, 是], [14, 5, 0, 0, 是], [14, 6, 18, 18, 是], [14, 7, 17, 17, 是], [14, 8, 16, 16, 是], [14, 9, 15, 15, 是], [14, 10, 14, 14, 是], [14, 11, 13, 13, 是], [14, 12, 12, 12, 是], [14, 13, 11, 11, 是], [14, 14, 10, 10, 是], [14, 15, 9, 9, 是], [14, 16, 8, 8, 是], [14, 17, 7, 7, 是], [14, 18, 6, 6, 是], [15, 0, 4, 4, 是], [15, 1, 3, 3, 是], [15, 2, 2, 2, 是], [15, 3, 1, 1, 是], [15, 4, 0, 0, 是], [15, 5, 18, 18, 是], [15, 6, 17, 17, 是], [15, 7, 16, 16, 是], [15, 8, 15, 15, 是], [15, 9, 14, 14, 是], [15, 10, 13, 13, 是], [15, 11, 12, 12, 是], [15, 12, 11, 11, 是], [15, 13, 10, 10, 是], [15, 14, 9, 9, 是], [15, 15, 8, 8, 是], [15, 16, 7, 7, 是], [15, 17, 6, 6, 是], [15, 18, 5, 5, 是], [16, 0, 3, 3, 是], [16, 1, 2, 2, 是], [16, 2, 1, 1, 是], [16, 3, 0, 0, 是], [16, 4, 18, 18, 是], [16, 5, 17, 17, 是], [16, 6, 16, 16, 是], [16, 7, 15, 15, 是], [16, 8, 14, 14, 是], [16, 9, 13, 13, 是], [16, 10, 12, 12, 是], [16, 11, 11, 11, 是], [16, 12, 10, 10, 是], [16, 13, 9, 9, 是], [16, 14, 8, 8, 是], [16, 15, 7, 7, 是], [16, 16, 6, 6, 是], [16, 17, 5, 5, 是], [16, 18, 4, 4, 是], [17, 0, 2, 2, 是], [17, 1, 1, 1, 是], [17, 2, 0, 0, 是], [17, 3, 18, 18, 是], [17, 4, 17, 17, 是], [17, 5, 16, 16, 是], [17, 6, 15, 15, 是], [17, 7, 14, 14, 是], [17, 8, 13, 13, 是], [17, 9, 12, 12, 是], [17, 10, 11, 11, 是], [17, 11, 10, 10, 是], [17, 12, 9, 9, 是], [17, 13, 8, 8, 是], [17, 14, 7, 7, 是], [17, 15, 6, 6, 是], [17, 16, 5, 5, 是], [17, 17, 4, 4, 是], [17, 18, 3, 3, 是], [18, 0, 1, 1, 是], [18, 1, 0, 0, 是], [18, 2, 18, 18, 是], [18, 3, 17, 17, 是], [18, 4, 16, 16, 是], [18, 5, 15, 15, 是], [18, 6, 14, 14, 是], [18, 7, 13, 13, 是], [18, 8, 12, 12, 是], [18, 9, 11, 11, 是], [18, 10, 10, 10, 是], [18, 11, 9, 9, 是], [18, 12, 8, 8, 是], [18, 13, 7, 7, 是], [18, 14, 6, 6, 是], [18, 15, 5, 5, 是], [18, 16, 4, 4, 是], [18, 17, 3, 3, 是], [18, 18, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(19, 19, [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 18, 18, 是], [0, 2, 17, 17, 是], [0, 3, 16, 16, 是], [0, 4, 15, 15, 是], [0, 5, 14, 14, 是], [0, 6, 13, 13, 是], [0, 7, 12, 12, 是], [0, 8, 11, 11, 是], [0, 9, 10, 10, 是], [0, 10, 9, 9, 是], [0, 11, 8, 8, 是], [0, 12, 7, 7, 是], [0, 13, 6, 6, 是], [0, 14, 5, 5, 是], [0, 15, 4, 4, 是], [0, 16, 3, 3, 是], [0, 17, 2, 2, 是], [0, 18, 1, 1, 是], [1, 0, 18, 18, 是], [1, 1, 17, 17, 是], [1, 2, 16, 16, 是], [1, 3, 15, 15, 是], [1, 4, 14, 14, 是], [1, 5, 13, 13, 是], [1, 6, 12, 12, 是], [1, 7, 11, 11, 是], [1, 8, 10, 10, 是], [1, 9, 9, 9, 是], [1, 10, 8, 8, 是], [1, 11, 7, 7, 是], [1, 12, 6, 6, 是], [1, 13, 5, 5, 是], [1, 14, 4, 4, 是], [1, 15, 3, 3, 是], [1, 16, 2, 2, 是], [1, 17, 1, 1, 是], [1, 18, 0, 0, 是], [2, 0, 17, 17, 是], [2, 1, 16, 16, 是], [2, 2, 15, 15, 是], [2, 3, 14, 14, 是], [2, 4, 13, 13, 是], [2, 5, 12, 12, 是], [2, 6, 11, 11, 是], [2, 7, 10, 10, 是], [2, 8, 9, 9, 是], [2, 9, 8, 8, 是], [2, 10, 7, 7, 是], [2, 11, 6, 6, 是], [2, 12, 5, 5, 是], [2, 13, 4, 4, 是], [2, 14, 3, 3, 是], [2, 15, 2, 2, 是], [2, 16, 1, 1, 是], [2, 17, 0, 0, 是], [2, 18, 18, 18, 是], [3, 0, 16, 16, 是], [3, 1, 15, 15, 是], [3, 2, 14, 14, 是], [3, 3, 13, 13, 是], [3, 4, 12, 12, 是], [3, 5, 11, 11, 是], [3, 6, 10, 10, 是], [3, 7, 9, 9, 是], [3, 8, 8, 8, 是], [3, 9, 7, 7, 是], [3, 10, 6, 6, 是], [3, 11, 5, 5, 是], [3, 12, 4, 4, 是], [3, 13, 3, 3, 是], [3, 14, 2, 2, 是], [3, 15, 1, 1, 是], [3, 16, 0, 0, 是], [3, 17, 18, 18, 是], [3, 18, 17, 17, 是], [4, 0, 15, 15, 是], [4, 1, 14, 14, 是], [4, 2, 13, 13, 是], [4, 3, 12, 12, 是], [4, 4, 11, 11, 是], [4, 5, 10, 10, 是], [4, 6, 9, 9, 是], [4, 7, 8, 8, 是], [4, 8, 7, 7, 是], [4, 9, 6, 6, 是], [4, 10, 5, 5, 是], [4, 11, 4, 4, 是], [4, 12, 3, 3, 是], [4, 13, 2, 2, 是], [4, 14, 1, 1, 是], [4, 15, 0, 0, 是], [4, 16, 18, 18, 是], [4, 17, 17, 17, 是], [4, 18, 16, 16, 是], [5, 0, 14, 14, 是], [5, 1, 13, 13, 是], [5, 2, 12, 12, 是], [5, 3, 11, 11, 是], [5, 4, 10, 10, 是], [5, 5, 9, 9, 是], [5, 6, 8, 8, 是], [5, 7, 7, 7, 是], [5, 8, 6, 6, 是], [5, 9, 5, 5, 是], [5, 10, 4, 4, 是], [5, 11, 3, 3, 是], [5, 12, 2, 2, 是], [5, 13, 1, 1, 是], [5, 14, 0, 0, 是], [5, 15, 18, 18, 是], [5, 16, 17, 17, 是], [5, 17, 16, 16, 是], [5, 18, 15, 15, 是], [6, 0, 13, 13, 是], [6, 1, 12, 12, 是], [6, 2, 11, 11, 是], [6, 3, 10, 10, 是], [6, 4, 9, 9, 是], [6, 5, 8, 8, 是], [6, 6, 7, 7, 是], [6, 7, 6, 6, 是], [6, 8, 5, 5, 是], [6, 9, 4, 4, 是], [6, 10, 3, 3, 是], [6, 11, 2, 2, 是], [6, 12, 1, 1, 是], [6, 13, 0, 0, 是], [6, 14, 18, 18, 是], [6, 15, 17, 17, 是], [6, 16, 16, 16, 是], [6, 17, 15, 15, 是], [6, 18, 14, 14, 是], [7, 0, 12, 12, 是], [7, 1, 11, 11, 是], [7, 2, 10, 10, 是], [7, 3, 9, 9, 是], [7, 4, 8, 8, 是], [7, 5, 7, 7, 是], [7, 6, 6, 6, 是], [7, 7, 5, 5, 是], [7, 8, 4, 4, 是], [7, 9, 3, 3, 是], [7, 10, 2, 2, 是], [7, 11, 1, 1, 是], [7, 12, 0, 0, 是], [7, 13, 18, 18, 是], [7, 14, 17, 17, 是], [7, 15, 16, 16, 是], [7, 16, 15, 15, 是], [7, 17, 14, 14, 是], [7, 18, 13, 13, 是], [8, 0, 11, 11, 是], [8, 1, 10, 10, 是], [8, 2, 9, 9, 是], [8, 3, 8, 8, 是], [8, 4, 7, 7, 是], [8, 5, 6, 6, 是], [8, 6, 5, 5, 是], [8, 7, 4, 4, 是], [8, 8, 3, 3, 是], [8, 9, 2, 2, 是], [8, 10, 1, 1, 是], [8, 11, 0, 0, 是], [8, 12, 18, 18, 是], [8, 13, 17, 17, 是], [8, 14, 16, 16, 是], [8, 15, 15, 15, 是], [8, 16, 14, 14, 是], [8, 17, 13, 13, 是], [8, 18, 12, 12, 是], [9, 0, 10, 10, 是], [9, 1, 9, 9, 是], [9, 2, 8, 8, 是], [9, 3, 7, 7, 是], [9, 4, 6, 6, 是], [9, 5, 5, 5, 是], [9, 6, 4, 4, 是], [9, 7, 3, 3, 是], [9, 8, 2, 2, 是], [9, 9, 1, 1, 是], [9, 10, 0, 0, 是], [9, 11, 18, 18, 是], [9, 12, 17, 17, 是], [9, 13, 16, 16, 是], [9, 14, 15, 15, 是], [9, 15, 14, 14, 是], [9, 16, 13, 13, 是], [9, 17, 12, 12, 是], [9, 18, 11, 11, 是], [10, 0, 9, 9, 是], [10, 1, 8, 8, 是], [10, 2, 7, 7, 是], [10, 3, 6, 6, 是], [10, 4, 5, 5, 是], [10, 5, 4, 4, 是], [10, 6, 3, 3, 是], [10, 7, 2, 2, 是], [10, 8, 1, 1, 是], [10, 9, 0, 0, 是], [10, 10, 18, 18, 是], [10, 11, 17, 17, 是], [10, 12, 16, 16, 是], [10, 13, 15, 15, 是], [10, 14, 14, 14, 是], [10, 15, 13, 13, 是], [10, 16, 12, 12, 是], [10, 17, 11, 11, 是], [10, 18, 10, 10, 是], [11, 0, 8, 8, 是], [11, 1, 7, 7, 是], [11, 2, 6, 6, 是], [11, 3, 5, 5, 是], [11, 4, 4, 4, 是], [11, 5, 3, 3, 是], [11, 6, 2, 2, 是], [11, 7, 1, 1, 是], [11, 8, 0, 0, 是], [11, 9, 18, 18, 是], [11, 10, 17, 17, 是], [11, 11, 16, 16, 是], [11, 12, 15, 15, 是], [11, 13, 14, 14, 是], [11, 14, 13, 13, 是], [11, 15, 12, 12, 是], [11, 16, 11, 11, 是], [11, 17, 10, 10, 是], [11, 18, 9, 9, 是], [12, 0, 7, 7, 是], [12, 1, 6, 6, 是], [12, 2, 5, 5, 是], [12, 3, 4, 4, 是], [12, 4, 3, 3, 是], [12, 5, 2, 2, 是], [12, 6, 1, 1, 是], [12, 7, 0, 0, 是], [12, 8, 18, 18, 是], [12, 9, 17, 17, 是], [12, 10, 16, 16, 是], [12, 11, 15, 15, 是], [12, 12, 14, 14, 是], [12, 13, 13, 13, 是], [12, 14, 12, 12, 是], [12, 15, 11, 11, 是], [12, 16, 10, 10, 是], [12, 17, 9, 9, 是], [12, 18, 8, 8, 是], [13, 0, 6, 6, 是], [13, 1, 5, 5, 是], [13, 2, 4, 4, 是], [13, 3, 3, 3, 是], [13, 4, 2, 2, 是], [13, 5, 1, 1, 是], [13, 6, 0, 0, 是], [13, 7, 18, 18, 是], [13, 8, 17, 17, 是], [13, 9, 16, 16, 是], [13, 10, 15, 15, 是], [13, 11, 14, 14, 是], [13, 12, 13, 13, 是], [13, 13, 12, 12, 是], [13, 14, 11, 11, 是], [13, 15, 10, 10, 是], [13, 16, 9, 9, 是], [13, 17, 8, 8, 是], [13, 18, 7, 7, 是], [14, 0, 5, 5, 是], [14, 1, 4, 4, 是], [14, 2, 3, 3, 是], [14, 3, 2, 2, 是], [14, 4, 1, 1, 是], [14, 5, 0, 0, 是], [14, 6, 18, 18, 是], [14, 7, 17, 17, 是], [14, 8, 16, 16, 是], [14, 9, 15, 15, 是], [14, 10, 14, 14, 是], [14, 11, 13, 13, 是], [14, 12, 12, 12, 是], [14, 13, 11, 11, 是], [14, 14, 10, 10, 是], [14, 15, 9, 9, 是], [14, 16, 8, 8, 是], [14, 17, 7, 7, 是], [14, 18, 6, 6, 是], [15, 0, 4, 4, 是], [15, 1, 3, 3, 是], [15, 2, 2, 2, 是], [15, 3, 1, 1, 是], [15, 4, 0, 0, 是], [15, 5, 18, 18, 是], [15, 6, 17, 17, 是], [15, 7, 16, 16, 是], [15, 8, 15, 15, 是], [15, 9, 14, 14, 是], [15, 10, 13, 13, 是], [15, 11, 12, 12, 是], [15, 12, 11, 11, 是], [15, 13, 10, 10, 是], [15, 14, 9, 9, 是], [15, 15, 8, 8, 是], [15, 16, 7, 7, 是], [15, 17, 6, 6, 是], [15, 18, 5, 5, 是], [16, 0, 3, 3, 是], [16, 1, 2, 2, 是], [16, 2, 1, 1, 是], [16, 3, 0, 0, 是], [16, 4, 18, 18, 是], [16, 5, 17, 17, 是], [16, 6, 16, 16, 是], [16, 7, 15, 15, 是], [16, 8, 14, 14, 是], [16, 9, 13, 13, 是], [16, 10, 12, 12, 是], [16, 11, 11, 11, 是], [16, 12, 10, 10, 是], [16, 13, 9, 9, 是], [16, 14, 8, 8, 是], [16, 15, 7, 7, 是], [16, 16, 6, 6, 是], [16, 17, 5, 5, 是], [16, 18, 4, 4, 是], [17, 0, 2, 2, 是], [17, 1, 1, 1, 是], [17, 2, 0, 0, 是], [17, 3, 18, 18, 是], [17, 4, 17, 17, 是], [17, 5, 16, 16, 是], [17, 6, 15, 15, 是], [17, 7, 14, 14, 是], [17, 8, 13, 13, 是], [17, 9, 12, 12, 是], [17, 10, 11, 11, 是], [17, 11, 10, 10, 是], [17, 12, 9, 9, 是], [17, 13, 8, 8, 是], [17, 14, 7, 7, 是], [17, 15, 6, 6, 是], [17, 16, 5, 5, 是], [17, 17, 4, 4, 是], [17, 18, 3, 3, 是], [18, 0, 1, 1, 是], [18, 1, 0, 0, 是], [18, 2, 18, 18, 是], [18, 3, 17, 17, 是], [18, 4, 16, 16, 是], [18, 5, 15, 15, 是], [18, 6, 14, 14, 是], [18, 7, 13, 13, 是], [18, 8, 12, 12, 是], [18, 9, 11, 11, 是], [18, 10, 10, 10, 是], [18, 11, 9, 9, 是], [18, 12, 8, 8, 是], [18, 13, 7, 7, 是], [18, 14, 6, 6, 是], [18, 15, 5, 5, 是], [18, 16, 4, 4, 是], [18, 17, 3, 3, 是], [18, 18, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141017,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],"target_identity":0,"mapping":[0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0018","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"mapping\":[0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 20, [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 19, 19, 是], [0, 2, 18, 18, 是], [0, 3, 17, 17, 是], [0, 4, 16, 16, 是], [0, 5, 15, 15, 是], [0, 6, 14, 14, 是], [0, 7, 13, 13, 是], [0, 8, 12, 12, 是], [0, 9, 11, 11, 是], [0, 10, 10, 10, 是], [0, 11, 9, 9, 是], [0, 12, 8, 8, 是], [0, 13, 7, 7, 是], [0, 14, 6, 6, 是], [0, 15, 5, 5, 是], [0, 16, 4, 4, 是], [0, 17, 3, 3, 是], [0, 18, 2, 2, 是], [0, 19, 1, 1, 是], [1, 0, 19, 19, 是], [1, 1, 18, 18, 是], [1, 2, 17, 17, 是], [1, 3, 16, 16, 是], [1, 4, 15, 15, 是], [1, 5, 14, 14, 是], [1, 6, 13, 13, 是], [1, 7, 12, 12, 是], [1, 8, 11, 11, 是], [1, 9, 10, 10, 是], [1, 10, 9, 9, 是], [1, 11, 8, 8, 是], [1, 12, 7, 7, 是], [1, 13, 6, 6, 是], [1, 14, 5, 5, 是], [1, 15, 4, 4, 是], [1, 16, 3, 3, 是], [1, 17, 2, 2, 是], [1, 18, 1, 1, 是], [1, 19, 0, 0, 是], [2, 0, 18, 18, 是], [2, 1, 17, 17, 是], [2, 2, 16, 16, 是], [2, 3, 15, 15, 是], [2, 4, 14, 14, 是], [2, 5, 13, 13, 是], [2, 6, 12, 12, 是], [2, 7, 11, 11, 是], [2, 8, 10, 10, 是], [2, 9, 9, 9, 是], [2, 10, 8, 8, 是], [2, 11, 7, 7, 是], [2, 12, 6, 6, 是], [2, 13, 5, 5, 是], [2, 14, 4, 4, 是], [2, 15, 3, 3, 是], [2, 16, 2, 2, 是], [2, 17, 1, 1, 是], [2, 18, 0, 0, 是], [2, 19, 19, 19, 是], [3, 0, 17, 17, 是], [3, 1, 16, 16, 是], [3, 2, 15, 15, 是], [3, 3, 14, 14, 是], [3, 4, 13, 13, 是], [3, 5, 12, 12, 是], [3, 6, 11, 11, 是], [3, 7, 10, 10, 是], [3, 8, 9, 9, 是], [3, 9, 8, 8, 是], [3, 10, 7, 7, 是], [3, 11, 6, 6, 是], [3, 12, 5, 5, 是], [3, 13, 4, 4, 是], [3, 14, 3, 3, 是], [3, 15, 2, 2, 是], [3, 16, 1, 1, 是], [3, 17, 0, 0, 是], [3, 18, 19, 19, 是], [3, 19, 18, 18, 是], [4, 0, 16, 16, 是], [4, 1, 15, 15, 是], [4, 2, 14, 14, 是], [4, 3, 13, 13, 是], [4, 4, 12, 12, 是], [4, 5, 11, 11, 是], [4, 6, 10, 10, 是], [4, 7, 9, 9, 是], [4, 8, 8, 8, 是], [4, 9, 7, 7, 是], [4, 10, 6, 6, 是], [4, 11, 5, 5, 是], [4, 12, 4, 4, 是], [4, 13, 3, 3, 是], [4, 14, 2, 2, 是], [4, 15, 1, 1, 是], [4, 16, 0, 0, 是], [4, 17, 19, 19, 是], [4, 18, 18, 18, 是], [4, 19, 17, 17, 是], [5, 0, 15, 15, 是], [5, 1, 14, 14, 是], [5, 2, 13, 13, 是], [5, 3, 12, 12, 是], [5, 4, 11, 11, 是], [5, 5, 10, 10, 是], [5, 6, 9, 9, 是], [5, 7, 8, 8, 是], [5, 8, 7, 7, 是], [5, 9, 6, 6, 是], [5, 10, 5, 5, 是], [5, 11, 4, 4, 是], [5, 12, 3, 3, 是], [5, 13, 2, 2, 是], [5, 14, 1, 1, 是], [5, 15, 0, 0, 是], [5, 16, 19, 19, 是], [5, 17, 18, 18, 是], [5, 18, 17, 17, 是], [5, 19, 16, 16, 是], [6, 0, 14, 14, 是], [6, 1, 13, 13, 是], [6, 2, 12, 12, 是], [6, 3, 11, 11, 是], [6, 4, 10, 10, 是], [6, 5, 9, 9, 是], [6, 6, 8, 8, 是], [6, 7, 7, 7, 是], [6, 8, 6, 6, 是], [6, 9, 5, 5, 是], [6, 10, 4, 4, 是], [6, 11, 3, 3, 是], [6, 12, 2, 2, 是], [6, 13, 1, 1, 是], [6, 14, 0, 0, 是], [6, 15, 19, 19, 是], [6, 16, 18, 18, 是], [6, 17, 17, 17, 是], [6, 18, 16, 16, 是], [6, 19, 15, 15, 是], [7, 0, 13, 13, 是], [7, 1, 12, 12, 是], [7, 2, 11, 11, 是], [7, 3, 10, 10, 是], [7, 4, 9, 9, 是], [7, 5, 8, 8, 是], [7, 6, 7, 7, 是], [7, 7, 6, 6, 是], [7, 8, 5, 5, 是], [7, 9, 4, 4, 是], [7, 10, 3, 3, 是], [7, 11, 2, 2, 是], [7, 12, 1, 1, 是], [7, 13, 0, 0, 是], [7, 14, 19, 19, 是], [7, 15, 18, 18, 是], [7, 16, 17, 17, 是], [7, 17, 16, 16, 是], [7, 18, 15, 15, 是], [7, 19, 14, 14, 是], [8, 0, 12, 12, 是], [8, 1, 11, 11, 是], [8, 2, 10, 10, 是], [8, 3, 9, 9, 是], [8, 4, 8, 8, 是], [8, 5, 7, 7, 是], [8, 6, 6, 6, 是], [8, 7, 5, 5, 是], [8, 8, 4, 4, 是], [8, 9, 3, 3, 是], [8, 10, 2, 2, 是], [8, 11, 1, 1, 是], [8, 12, 0, 0, 是], [8, 13, 19, 19, 是], [8, 14, 18, 18, 是], [8, 15, 17, 17, 是], [8, 16, 16, 16, 是], [8, 17, 15, 15, 是], [8, 18, 14, 14, 是], [8, 19, 13, 13, 是], [9, 0, 11, 11, 是], [9, 1, 10, 10, 是], [9, 2, 9, 9, 是], [9, 3, 8, 8, 是], [9, 4, 7, 7, 是], [9, 5, 6, 6, 是], [9, 6, 5, 5, 是], [9, 7, 4, 4, 是], [9, 8, 3, 3, 是], [9, 9, 2, 2, 是], [9, 10, 1, 1, 是], [9, 11, 0, 0, 是], [9, 12, 19, 19, 是], [9, 13, 18, 18, 是], [9, 14, 17, 17, 是], [9, 15, 16, 16, 是], [9, 16, 15, 15, 是], [9, 17, 14, 14, 是], [9, 18, 13, 13, 是], [9, 19, 12, 12, 是], [10, 0, 10, 10, 是], [10, 1, 9, 9, 是], [10, 2, 8, 8, 是], [10, 3, 7, 7, 是], [10, 4, 6, 6, 是], [10, 5, 5, 5, 是], [10, 6, 4, 4, 是], [10, 7, 3, 3, 是], [10, 8, 2, 2, 是], [10, 9, 1, 1, 是], [10, 10, 0, 0, 是], [10, 11, 19, 19, 是], [10, 12, 18, 18, 是], [10, 13, 17, 17, 是], [10, 14, 16, 16, 是], [10, 15, 15, 15, 是], [10, 16, 14, 14, 是], [10, 17, 13, 13, 是], [10, 18, 12, 12, 是], [10, 19, 11, 11, 是], [11, 0, 9, 9, 是], [11, 1, 8, 8, 是], [11, 2, 7, 7, 是], [11, 3, 6, 6, 是], [11, 4, 5, 5, 是], [11, 5, 4, 4, 是], [11, 6, 3, 3, 是], [11, 7, 2, 2, 是], [11, 8, 1, 1, 是], [11, 9, 0, 0, 是], [11, 10, 19, 19, 是], [11, 11, 18, 18, 是], [11, 12, 17, 17, 是], [11, 13, 16, 16, 是], [11, 14, 15, 15, 是], [11, 15, 14, 14, 是], [11, 16, 13, 13, 是], [11, 17, 12, 12, 是], [11, 18, 11, 11, 是], [11, 19, 10, 10, 是], [12, 0, 8, 8, 是], [12, 1, 7, 7, 是], [12, 2, 6, 6, 是], [12, 3, 5, 5, 是], [12, 4, 4, 4, 是], [12, 5, 3, 3, 是], [12, 6, 2, 2, 是], [12, 7, 1, 1, 是], [12, 8, 0, 0, 是], [12, 9, 19, 19, 是], [12, 10, 18, 18, 是], [12, 11, 17, 17, 是], [12, 12, 16, 16, 是], [12, 13, 15, 15, 是], [12, 14, 14, 14, 是], [12, 15, 13, 13, 是], [12, 16, 12, 12, 是], [12, 17, 11, 11, 是], [12, 18, 10, 10, 是], [12, 19, 9, 9, 是], [13, 0, 7, 7, 是], [13, 1, 6, 6, 是], [13, 2, 5, 5, 是], [13, 3, 4, 4, 是], [13, 4, 3, 3, 是], [13, 5, 2, 2, 是], [13, 6, 1, 1, 是], [13, 7, 0, 0, 是], [13, 8, 19, 19, 是], [13, 9, 18, 18, 是], [13, 10, 17, 17, 是], [13, 11, 16, 16, 是], [13, 12, 15, 15, 是], [13, 13, 14, 14, 是], [13, 14, 13, 13, 是], [13, 15, 12, 12, 是], [13, 16, 11, 11, 是], [13, 17, 10, 10, 是], [13, 18, 9, 9, 是], [13, 19, 8, 8, 是], [14, 0, 6, 6, 是], [14, 1, 5, 5, 是], [14, 2, 4, 4, 是], [14, 3, 3, 3, 是], [14, 4, 2, 2, 是], [14, 5, 1, 1, 是], [14, 6, 0, 0, 是], [14, 7, 19, 19, 是], [14, 8, 18, 18, 是], [14, 9, 17, 17, 是], [14, 10, 16, 16, 是], [14, 11, 15, 15, 是], [14, 12, 14, 14, 是], [14, 13, 13, 13, 是], [14, 14, 12, 12, 是], [14, 15, 11, 11, 是], [14, 16, 10, 10, 是], [14, 17, 9, 9, 是], [14, 18, 8, 8, 是], [14, 19, 7, 7, 是], [15, 0, 5, 5, 是], [15, 1, 4, 4, 是], [15, 2, 3, 3, 是], [15, 3, 2, 2, 是], [15, 4, 1, 1, 是], [15, 5, 0, 0, 是], [15, 6, 19, 19, 是], [15, 7, 18, 18, 是], [15, 8, 17, 17, 是], [15, 9, 16, 16, 是], [15, 10, 15, 15, 是], [15, 11, 14, 14, 是], [15, 12, 13, 13, 是], [15, 13, 12, 12, 是], [15, 14, 11, 11, 是], [15, 15, 10, 10, 是], [15, 16, 9, 9, 是], [15, 17, 8, 8, 是], [15, 18, 7, 7, 是], [15, 19, 6, 6, 是], [16, 0, 4, 4, 是], [16, 1, 3, 3, 是], [16, 2, 2, 2, 是], [16, 3, 1, 1, 是], [16, 4, 0, 0, 是], [16, 5, 19, 19, 是], [16, 6, 18, 18, 是], [16, 7, 17, 17, 是], [16, 8, 16, 16, 是], [16, 9, 15, 15, 是], [16, 10, 14, 14, 是], [16, 11, 13, 13, 是], [16, 12, 12, 12, 是], [16, 13, 11, 11, 是], [16, 14, 10, 10, 是], [16, 15, 9, 9, 是], [16, 16, 8, 8, 是], [16, 17, 7, 7, 是], [16, 18, 6, 6, 是], [16, 19, 5, 5, 是], [17, 0, 3, 3, 是], [17, 1, 2, 2, 是], [17, 2, 1, 1, 是], [17, 3, 0, 0, 是], [17, 4, 19, 19, 是], [17, 5, 18, 18, 是], [17, 6, 17, 17, 是], [17, 7, 16, 16, 是], [17, 8, 15, 15, 是], [17, 9, 14, 14, 是], [17, 10, 13, 13, 是], [17, 11, 12, 12, 是], [17, 12, 11, 11, 是], [17, 13, 10, 10, 是], [17, 14, 9, 9, 是], [17, 15, 8, 8, 是], [17, 16, 7, 7, 是], [17, 17, 6, 6, 是], [17, 18, 5, 5, 是], [17, 19, 4, 4, 是], [18, 0, 2, 2, 是], [18, 1, 1, 1, 是], [18, 2, 0, 0, 是], [18, 3, 19, 19, 是], [18, 4, 18, 18, 是], [18, 5, 17, 17, 是], [18, 6, 16, 16, 是], [18, 7, 15, 15, 是], [18, 8, 14, 14, 是], [18, 9, 13, 13, 是], [18, 10, 12, 12, 是], [18, 11, 11, 11, 是], [18, 12, 10, 10, 是], [18, 13, 9, 9, 是], [18, 14, 8, 8, 是], [18, 15, 7, 7, 是], [18, 16, 6, 6, 是], [18, 17, 5, 5, 是], [18, 18, 4, 4, 是], [18, 19, 3, 3, 是], [19, 0, 1, 1, 是], [19, 1, 0, 0, 是], [19, 2, 19, 19, 是], [19, 3, 18, 18, 是], [19, 4, 17, 17, 是], [19, 5, 16, 16, 是], [19, 6, 15, 15, 是], [19, 7, 14, 14, 是], [19, 8, 13, 13, 是], [19, 9, 12, 12, 是], [19, 10, 11, 11, 是], [19, 11, 10, 10, 是], [19, 12, 9, 9, 是], [19, 13, 8, 8, 是], [19, 14, 7, 7, 是], [19, 15, 6, 6, 是], [19, 16, 5, 5, 是], [19, 17, 4, 4, 是], [19, 18, 3, 3, 是], [19, 19, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(20, 20, [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 19, 19, 是], [0, 2, 18, 18, 是], [0, 3, 17, 17, 是], [0, 4, 16, 16, 是], [0, 5, 15, 15, 是], [0, 6, 14, 14, 是], [0, 7, 13, 13, 是], [0, 8, 12, 12, 是], [0, 9, 11, 11, 是], [0, 10, 10, 10, 是], [0, 11, 9, 9, 是], [0, 12, 8, 8, 是], [0, 13, 7, 7, 是], [0, 14, 6, 6, 是], [0, 15, 5, 5, 是], [0, 16, 4, 4, 是], [0, 17, 3, 3, 是], [0, 18, 2, 2, 是], [0, 19, 1, 1, 是], [1, 0, 19, 19, 是], [1, 1, 18, 18, 是], [1, 2, 17, 17, 是], [1, 3, 16, 16, 是], [1, 4, 15, 15, 是], [1, 5, 14, 14, 是], [1, 6, 13, 13, 是], [1, 7, 12, 12, 是], [1, 8, 11, 11, 是], [1, 9, 10, 10, 是], [1, 10, 9, 9, 是], [1, 11, 8, 8, 是], [1, 12, 7, 7, 是], [1, 13, 6, 6, 是], [1, 14, 5, 5, 是], [1, 15, 4, 4, 是], [1, 16, 3, 3, 是], [1, 17, 2, 2, 是], [1, 18, 1, 1, 是], [1, 19, 0, 0, 是], [2, 0, 18, 18, 是], [2, 1, 17, 17, 是], [2, 2, 16, 16, 是], [2, 3, 15, 15, 是], [2, 4, 14, 14, 是], [2, 5, 13, 13, 是], [2, 6, 12, 12, 是], [2, 7, 11, 11, 是], [2, 8, 10, 10, 是], [2, 9, 9, 9, 是], [2, 10, 8, 8, 是], [2, 11, 7, 7, 是], [2, 12, 6, 6, 是], [2, 13, 5, 5, 是], [2, 14, 4, 4, 是], [2, 15, 3, 3, 是], [2, 16, 2, 2, 是], [2, 17, 1, 1, 是], [2, 18, 0, 0, 是], [2, 19, 19, 19, 是], [3, 0, 17, 17, 是], [3, 1, 16, 16, 是], [3, 2, 15, 15, 是], [3, 3, 14, 14, 是], [3, 4, 13, 13, 是], [3, 5, 12, 12, 是], [3, 6, 11, 11, 是], [3, 7, 10, 10, 是], [3, 8, 9, 9, 是], [3, 9, 8, 8, 是], [3, 10, 7, 7, 是], [3, 11, 6, 6, 是], [3, 12, 5, 5, 是], [3, 13, 4, 4, 是], [3, 14, 3, 3, 是], [3, 15, 2, 2, 是], [3, 16, 1, 1, 是], [3, 17, 0, 0, 是], [3, 18, 19, 19, 是], [3, 19, 18, 18, 是], [4, 0, 16, 16, 是], [4, 1, 15, 15, 是], [4, 2, 14, 14, 是], [4, 3, 13, 13, 是], [4, 4, 12, 12, 是], [4, 5, 11, 11, 是], [4, 6, 10, 10, 是], [4, 7, 9, 9, 是], [4, 8, 8, 8, 是], [4, 9, 7, 7, 是], [4, 10, 6, 6, 是], [4, 11, 5, 5, 是], [4, 12, 4, 4, 是], [4, 13, 3, 3, 是], [4, 14, 2, 2, 是], [4, 15, 1, 1, 是], [4, 16, 0, 0, 是], [4, 17, 19, 19, 是], [4, 18, 18, 18, 是], [4, 19, 17, 17, 是], [5, 0, 15, 15, 是], [5, 1, 14, 14, 是], [5, 2, 13, 13, 是], [5, 3, 12, 12, 是], [5, 4, 11, 11, 是], [5, 5, 10, 10, 是], [5, 6, 9, 9, 是], [5, 7, 8, 8, 是], [5, 8, 7, 7, 是], [5, 9, 6, 6, 是], [5, 10, 5, 5, 是], [5, 11, 4, 4, 是], [5, 12, 3, 3, 是], [5, 13, 2, 2, 是], [5, 14, 1, 1, 是], [5, 15, 0, 0, 是], [5, 16, 19, 19, 是], [5, 17, 18, 18, 是], [5, 18, 17, 17, 是], [5, 19, 16, 16, 是], [6, 0, 14, 14, 是], [6, 1, 13, 13, 是], [6, 2, 12, 12, 是], [6, 3, 11, 11, 是], [6, 4, 10, 10, 是], [6, 5, 9, 9, 是], [6, 6, 8, 8, 是], [6, 7, 7, 7, 是], [6, 8, 6, 6, 是], [6, 9, 5, 5, 是], [6, 10, 4, 4, 是], [6, 11, 3, 3, 是], [6, 12, 2, 2, 是], [6, 13, 1, 1, 是], [6, 14, 0, 0, 是], [6, 15, 19, 19, 是], [6, 16, 18, 18, 是], [6, 17, 17, 17, 是], [6, 18, 16, 16, 是], [6, 19, 15, 15, 是], [7, 0, 13, 13, 是], [7, 1, 12, 12, 是], [7, 2, 11, 11, 是], [7, 3, 10, 10, 是], [7, 4, 9, 9, 是], [7, 5, 8, 8, 是], [7, 6, 7, 7, 是], [7, 7, 6, 6, 是], [7, 8, 5, 5, 是], [7, 9, 4, 4, 是], [7, 10, 3, 3, 是], [7, 11, 2, 2, 是], [7, 12, 1, 1, 是], [7, 13, 0, 0, 是], [7, 14, 19, 19, 是], [7, 15, 18, 18, 是], [7, 16, 17, 17, 是], [7, 17, 16, 16, 是], [7, 18, 15, 15, 是], [7, 19, 14, 14, 是], [8, 0, 12, 12, 是], [8, 1, 11, 11, 是], [8, 2, 10, 10, 是], [8, 3, 9, 9, 是], [8, 4, 8, 8, 是], [8, 5, 7, 7, 是], [8, 6, 6, 6, 是], [8, 7, 5, 5, 是], [8, 8, 4, 4, 是], [8, 9, 3, 3, 是], [8, 10, 2, 2, 是], [8, 11, 1, 1, 是], [8, 12, 0, 0, 是], [8, 13, 19, 19, 是], [8, 14, 18, 18, 是], [8, 15, 17, 17, 是], [8, 16, 16, 16, 是], [8, 17, 15, 15, 是], [8, 18, 14, 14, 是], [8, 19, 13, 13, 是], [9, 0, 11, 11, 是], [9, 1, 10, 10, 是], [9, 2, 9, 9, 是], [9, 3, 8, 8, 是], [9, 4, 7, 7, 是], [9, 5, 6, 6, 是], [9, 6, 5, 5, 是], [9, 7, 4, 4, 是], [9, 8, 3, 3, 是], [9, 9, 2, 2, 是], [9, 10, 1, 1, 是], [9, 11, 0, 0, 是], [9, 12, 19, 19, 是], [9, 13, 18, 18, 是], [9, 14, 17, 17, 是], [9, 15, 16, 16, 是], [9, 16, 15, 15, 是], [9, 17, 14, 14, 是], [9, 18, 13, 13, 是], [9, 19, 12, 12, 是], [10, 0, 10, 10, 是], [10, 1, 9, 9, 是], [10, 2, 8, 8, 是], [10, 3, 7, 7, 是], [10, 4, 6, 6, 是], [10, 5, 5, 5, 是], [10, 6, 4, 4, 是], [10, 7, 3, 3, 是], [10, 8, 2, 2, 是], [10, 9, 1, 1, 是], [10, 10, 0, 0, 是], [10, 11, 19, 19, 是], [10, 12, 18, 18, 是], [10, 13, 17, 17, 是], [10, 14, 16, 16, 是], [10, 15, 15, 15, 是], [10, 16, 14, 14, 是], [10, 17, 13, 13, 是], [10, 18, 12, 12, 是], [10, 19, 11, 11, 是], [11, 0, 9, 9, 是], [11, 1, 8, 8, 是], [11, 2, 7, 7, 是], [11, 3, 6, 6, 是], [11, 4, 5, 5, 是], [11, 5, 4, 4, 是], [11, 6, 3, 3, 是], [11, 7, 2, 2, 是], [11, 8, 1, 1, 是], [11, 9, 0, 0, 是], [11, 10, 19, 19, 是], [11, 11, 18, 18, 是], [11, 12, 17, 17, 是], [11, 13, 16, 16, 是], [11, 14, 15, 15, 是], [11, 15, 14, 14, 是], [11, 16, 13, 13, 是], [11, 17, 12, 12, 是], [11, 18, 11, 11, 是], [11, 19, 10, 10, 是], [12, 0, 8, 8, 是], [12, 1, 7, 7, 是], [12, 2, 6, 6, 是], [12, 3, 5, 5, 是], [12, 4, 4, 4, 是], [12, 5, 3, 3, 是], [12, 6, 2, 2, 是], [12, 7, 1, 1, 是], [12, 8, 0, 0, 是], [12, 9, 19, 19, 是], [12, 10, 18, 18, 是], [12, 11, 17, 17, 是], [12, 12, 16, 16, 是], [12, 13, 15, 15, 是], [12, 14, 14, 14, 是], [12, 15, 13, 13, 是], [12, 16, 12, 12, 是], [12, 17, 11, 11, 是], [12, 18, 10, 10, 是], [12, 19, 9, 9, 是], [13, 0, 7, 7, 是], [13, 1, 6, 6, 是], [13, 2, 5, 5, 是], [13, 3, 4, 4, 是], [13, 4, 3, 3, 是], [13, 5, 2, 2, 是], [13, 6, 1, 1, 是], [13, 7, 0, 0, 是], [13, 8, 19, 19, 是], [13, 9, 18, 18, 是], [13, 10, 17, 17, 是], [13, 11, 16, 16, 是], [13, 12, 15, 15, 是], [13, 13, 14, 14, 是], [13, 14, 13, 13, 是], [13, 15, 12, 12, 是], [13, 16, 11, 11, 是], [13, 17, 10, 10, 是], [13, 18, 9, 9, 是], [13, 19, 8, 8, 是], [14, 0, 6, 6, 是], [14, 1, 5, 5, 是], [14, 2, 4, 4, 是], [14, 3, 3, 3, 是], [14, 4, 2, 2, 是], [14, 5, 1, 1, 是], [14, 6, 0, 0, 是], [14, 7, 19, 19, 是], [14, 8, 18, 18, 是], [14, 9, 17, 17, 是], [14, 10, 16, 16, 是], [14, 11, 15, 15, 是], [14, 12, 14, 14, 是], [14, 13, 13, 13, 是], [14, 14, 12, 12, 是], [14, 15, 11, 11, 是], [14, 16, 10, 10, 是], [14, 17, 9, 9, 是], [14, 18, 8, 8, 是], [14, 19, 7, 7, 是], [15, 0, 5, 5, 是], [15, 1, 4, 4, 是], [15, 2, 3, 3, 是], [15, 3, 2, 2, 是], [15, 4, 1, 1, 是], [15, 5, 0, 0, 是], [15, 6, 19, 19, 是], [15, 7, 18, 18, 是], [15, 8, 17, 17, 是], [15, 9, 16, 16, 是], [15, 10, 15, 15, 是], [15, 11, 14, 14, 是], [15, 12, 13, 13, 是], [15, 13, 12, 12, 是], [15, 14, 11, 11, 是], [15, 15, 10, 10, 是], [15, 16, 9, 9, 是], [15, 17, 8, 8, 是], [15, 18, 7, 7, 是], [15, 19, 6, 6, 是], [16, 0, 4, 4, 是], [16, 1, 3, 3, 是], [16, 2, 2, 2, 是], [16, 3, 1, 1, 是], [16, 4, 0, 0, 是], [16, 5, 19, 19, 是], [16, 6, 18, 18, 是], [16, 7, 17, 17, 是], [16, 8, 16, 16, 是], [16, 9, 15, 15, 是], [16, 10, 14, 14, 是], [16, 11, 13, 13, 是], [16, 12, 12, 12, 是], [16, 13, 11, 11, 是], [16, 14, 10, 10, 是], [16, 15, 9, 9, 是], [16, 16, 8, 8, 是], [16, 17, 7, 7, 是], [16, 18, 6, 6, 是], [16, 19, 5, 5, 是], [17, 0, 3, 3, 是], [17, 1, 2, 2, 是], [17, 2, 1, 1, 是], [17, 3, 0, 0, 是], [17, 4, 19, 19, 是], [17, 5, 18, 18, 是], [17, 6, 17, 17, 是], [17, 7, 16, 16, 是], [17, 8, 15, 15, 是], [17, 9, 14, 14, 是], [17, 10, 13, 13, 是], [17, 11, 12, 12, 是], [17, 12, 11, 11, 是], [17, 13, 10, 10, 是], [17, 14, 9, 9, 是], [17, 15, 8, 8, 是], [17, 16, 7, 7, 是], [17, 17, 6, 6, 是], [17, 18, 5, 5, 是], [17, 19, 4, 4, 是], [18, 0, 2, 2, 是], [18, 1, 1, 1, 是], [18, 2, 0, 0, 是], [18, 3, 19, 19, 是], [18, 4, 18, 18, 是], [18, 5, 17, 17, 是], [18, 6, 16, 16, 是], [18, 7, 15, 15, 是], [18, 8, 14, 14, 是], [18, 9, 13, 13, 是], [18, 10, 12, 12, 是], [18, 11, 11, 11, 是], [18, 12, 10, 10, 是], [18, 13, 9, 9, 是], [18, 14, 8, 8, 是], [18, 15, 7, 7, 是], [18, 16, 6, 6, 是], [18, 17, 5, 5, 是], [18, 18, 4, 4, 是], [18, 19, 3, 3, 是], [19, 0, 1, 1, 是], [19, 1, 0, 0, 是], [19, 2, 19, 19, 是], [19, 3, 18, 18, 是], [19, 4, 17, 17, 是], [19, 5, 16, 16, 是], [19, 6, 15, 15, 是], [19, 7, 14, 14, 是], [19, 8, 13, 13, 是], [19, 9, 12, 12, 是], [19, 10, 11, 11, 是], [19, 11, 10, 10, 是], [19, 12, 9, 9, 是], [19, 13, 8, 8, 是], [19, 14, 7, 7, 是], [19, 15, 6, 6, 是], [19, 16, 5, 5, 是], [19, 17, 4, 4, 是], [19, 18, 3, 3, 是], [19, 19, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141018,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],"target_identity":0,"mapping":[0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0019","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],\"target_identity\":0,\"mapping\":[0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 21, [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 20, 20, 是], [0, 2, 19, 19, 是], [0, 3, 18, 18, 是], [0, 4, 17, 17, 是], [0, 5, 16, 16, 是], [0, 6, 15, 15, 是], [0, 7, 14, 14, 是], [0, 8, 13, 13, 是], [0, 9, 12, 12, 是], [0, 10, 11, 11, 是], [0, 11, 10, 10, 是], [0, 12, 9, 9, 是], [0, 13, 8, 8, 是], [0, 14, 7, 7, 是], [0, 15, 6, 6, 是], [0, 16, 5, 5, 是], [0, 17, 4, 4, 是], [0, 18, 3, 3, 是], [0, 19, 2, 2, 是], [0, 20, 1, 1, 是], [1, 0, 20, 20, 是], [1, 1, 19, 19, 是], [1, 2, 18, 18, 是], [1, 3, 17, 17, 是], [1, 4, 16, 16, 是], [1, 5, 15, 15, 是], [1, 6, 14, 14, 是], [1, 7, 13, 13, 是], [1, 8, 12, 12, 是], [1, 9, 11, 11, 是], [1, 10, 10, 10, 是], [1, 11, 9, 9, 是], [1, 12, 8, 8, 是], [1, 13, 7, 7, 是], [1, 14, 6, 6, 是], [1, 15, 5, 5, 是], [1, 16, 4, 4, 是], [1, 17, 3, 3, 是], [1, 18, 2, 2, 是], [1, 19, 1, 1, 是], [1, 20, 0, 0, 是], [2, 0, 19, 19, 是], [2, 1, 18, 18, 是], [2, 2, 17, 17, 是], [2, 3, 16, 16, 是], [2, 4, 15, 15, 是], [2, 5, 14, 14, 是], [2, 6, 13, 13, 是], [2, 7, 12, 12, 是], [2, 8, 11, 11, 是], [2, 9, 10, 10, 是], [2, 10, 9, 9, 是], [2, 11, 8, 8, 是], [2, 12, 7, 7, 是], [2, 13, 6, 6, 是], [2, 14, 5, 5, 是], [2, 15, 4, 4, 是], [2, 16, 3, 3, 是], [2, 17, 2, 2, 是], [2, 18, 1, 1, 是], [2, 19, 0, 0, 是], [2, 20, 20, 20, 是], [3, 0, 18, 18, 是], [3, 1, 17, 17, 是], [3, 2, 16, 16, 是], [3, 3, 15, 15, 是], [3, 4, 14, 14, 是], [3, 5, 13, 13, 是], [3, 6, 12, 12, 是], [3, 7, 11, 11, 是], [3, 8, 10, 10, 是], [3, 9, 9, 9, 是], [3, 10, 8, 8, 是], [3, 11, 7, 7, 是], [3, 12, 6, 6, 是], [3, 13, 5, 5, 是], [3, 14, 4, 4, 是], [3, 15, 3, 3, 是], [3, 16, 2, 2, 是], [3, 17, 1, 1, 是], [3, 18, 0, 0, 是], [3, 19, 20, 20, 是], [3, 20, 19, 19, 是], [4, 0, 17, 17, 是], [4, 1, 16, 16, 是], [4, 2, 15, 15, 是], [4, 3, 14, 14, 是], [4, 4, 13, 13, 是], [4, 5, 12, 12, 是], [4, 6, 11, 11, 是], [4, 7, 10, 10, 是], [4, 8, 9, 9, 是], [4, 9, 8, 8, 是], [4, 10, 7, 7, 是], [4, 11, 6, 6, 是], [4, 12, 5, 5, 是], [4, 13, 4, 4, 是], [4, 14, 3, 3, 是], [4, 15, 2, 2, 是], [4, 16, 1, 1, 是], [4, 17, 0, 0, 是], [4, 18, 20, 20, 是], [4, 19, 19, 19, 是], [4, 20, 18, 18, 是], [5, 0, 16, 16, 是], [5, 1, 15, 15, 是], [5, 2, 14, 14, 是], [5, 3, 13, 13, 是], [5, 4, 12, 12, 是], [5, 5, 11, 11, 是], [5, 6, 10, 10, 是], [5, 7, 9, 9, 是], [5, 8, 8, 8, 是], [5, 9, 7, 7, 是], [5, 10, 6, 6, 是], [5, 11, 5, 5, 是], [5, 12, 4, 4, 是], [5, 13, 3, 3, 是], [5, 14, 2, 2, 是], [5, 15, 1, 1, 是], [5, 16, 0, 0, 是], [5, 17, 20, 20, 是], [5, 18, 19, 19, 是], [5, 19, 18, 18, 是], [5, 20, 17, 17, 是], [6, 0, 15, 15, 是], [6, 1, 14, 14, 是], [6, 2, 13, 13, 是], [6, 3, 12, 12, 是], [6, 4, 11, 11, 是], [6, 5, 10, 10, 是], [6, 6, 9, 9, 是], [6, 7, 8, 8, 是], [6, 8, 7, 7, 是], [6, 9, 6, 6, 是], [6, 10, 5, 5, 是], [6, 11, 4, 4, 是], [6, 12, 3, 3, 是], [6, 13, 2, 2, 是], [6, 14, 1, 1, 是], [6, 15, 0, 0, 是], [6, 16, 20, 20, 是], [6, 17, 19, 19, 是], [6, 18, 18, 18, 是], [6, 19, 17, 17, 是], [6, 20, 16, 16, 是], [7, 0, 14, 14, 是], [7, 1, 13, 13, 是], [7, 2, 12, 12, 是], [7, 3, 11, 11, 是], [7, 4, 10, 10, 是], [7, 5, 9, 9, 是], [7, 6, 8, 8, 是], [7, 7, 7, 7, 是], [7, 8, 6, 6, 是], [7, 9, 5, 5, 是], [7, 10, 4, 4, 是], [7, 11, 3, 3, 是], [7, 12, 2, 2, 是], [7, 13, 1, 1, 是], [7, 14, 0, 0, 是], [7, 15, 20, 20, 是], [7, 16, 19, 19, 是], [7, 17, 18, 18, 是], [7, 18, 17, 17, 是], [7, 19, 16, 16, 是], [7, 20, 15, 15, 是], [8, 0, 13, 13, 是], [8, 1, 12, 12, 是], [8, 2, 11, 11, 是], [8, 3, 10, 10, 是], [8, 4, 9, 9, 是], [8, 5, 8, 8, 是], [8, 6, 7, 7, 是], [8, 7, 6, 6, 是], [8, 8, 5, 5, 是], [8, 9, 4, 4, 是], [8, 10, 3, 3, 是], [8, 11, 2, 2, 是], [8, 12, 1, 1, 是], [8, 13, 0, 0, 是], [8, 14, 20, 20, 是], [8, 15, 19, 19, 是], [8, 16, 18, 18, 是], [8, 17, 17, 17, 是], [8, 18, 16, 16, 是], [8, 19, 15, 15, 是], [8, 20, 14, 14, 是], [9, 0, 12, 12, 是], [9, 1, 11, 11, 是], [9, 2, 10, 10, 是], [9, 3, 9, 9, 是], [9, 4, 8, 8, 是], [9, 5, 7, 7, 是], [9, 6, 6, 6, 是], [9, 7, 5, 5, 是], [9, 8, 4, 4, 是], [9, 9, 3, 3, 是], [9, 10, 2, 2, 是], [9, 11, 1, 1, 是], [9, 12, 0, 0, 是], [9, 13, 20, 20, 是], [9, 14, 19, 19, 是], [9, 15, 18, 18, 是], [9, 16, 17, 17, 是], [9, 17, 16, 16, 是], [9, 18, 15, 15, 是], [9, 19, 14, 14, 是], [9, 20, 13, 13, 是], [10, 0, 11, 11, 是], [10, 1, 10, 10, 是], [10, 2, 9, 9, 是], [10, 3, 8, 8, 是], [10, 4, 7, 7, 是], [10, 5, 6, 6, 是], [10, 6, 5, 5, 是], [10, 7, 4, 4, 是], [10, 8, 3, 3, 是], [10, 9, 2, 2, 是], [10, 10, 1, 1, 是], [10, 11, 0, 0, 是], [10, 12, 20, 20, 是], [10, 13, 19, 19, 是], [10, 14, 18, 18, 是], [10, 15, 17, 17, 是], [10, 16, 16, 16, 是], [10, 17, 15, 15, 是], [10, 18, 14, 14, 是], [10, 19, 13, 13, 是], [10, 20, 12, 12, 是], [11, 0, 10, 10, 是], [11, 1, 9, 9, 是], [11, 2, 8, 8, 是], [11, 3, 7, 7, 是], [11, 4, 6, 6, 是], [11, 5, 5, 5, 是], [11, 6, 4, 4, 是], [11, 7, 3, 3, 是], [11, 8, 2, 2, 是], [11, 9, 1, 1, 是], [11, 10, 0, 0, 是], [11, 11, 20, 20, 是], [11, 12, 19, 19, 是], [11, 13, 18, 18, 是], [11, 14, 17, 17, 是], [11, 15, 16, 16, 是], [11, 16, 15, 15, 是], [11, 17, 14, 14, 是], [11, 18, 13, 13, 是], [11, 19, 12, 12, 是], [11, 20, 11, 11, 是], [12, 0, 9, 9, 是], [12, 1, 8, 8, 是], [12, 2, 7, 7, 是], [12, 3, 6, 6, 是], [12, 4, 5, 5, 是], [12, 5, 4, 4, 是], [12, 6, 3, 3, 是], [12, 7, 2, 2, 是], [12, 8, 1, 1, 是], [12, 9, 0, 0, 是], [12, 10, 20, 20, 是], [12, 11, 19, 19, 是], [12, 12, 18, 18, 是], [12, 13, 17, 17, 是], [12, 14, 16, 16, 是], [12, 15, 15, 15, 是], [12, 16, 14, 14, 是], [12, 17, 13, 13, 是], [12, 18, 12, 12, 是], [12, 19, 11, 11, 是], [12, 20, 10, 10, 是], [13, 0, 8, 8, 是], [13, 1, 7, 7, 是], [13, 2, 6, 6, 是], [13, 3, 5, 5, 是], [13, 4, 4, 4, 是], [13, 5, 3, 3, 是], [13, 6, 2, 2, 是], [13, 7, 1, 1, 是], [13, 8, 0, 0, 是], [13, 9, 20, 20, 是], [13, 10, 19, 19, 是], [13, 11, 18, 18, 是], [13, 12, 17, 17, 是], [13, 13, 16, 16, 是], [13, 14, 15, 15, 是], [13, 15, 14, 14, 是], [13, 16, 13, 13, 是], [13, 17, 12, 12, 是], [13, 18, 11, 11, 是], [13, 19, 10, 10, 是], [13, 20, 9, 9, 是], [14, 0, 7, 7, 是], [14, 1, 6, 6, 是], [14, 2, 5, 5, 是], [14, 3, 4, 4, 是], [14, 4, 3, 3, 是], [14, 5, 2, 2, 是], [14, 6, 1, 1, 是], [14, 7, 0, 0, 是], [14, 8, 20, 20, 是], [14, 9, 19, 19, 是], [14, 10, 18, 18, 是], [14, 11, 17, 17, 是], [14, 12, 16, 16, 是], [14, 13, 15, 15, 是], [14, 14, 14, 14, 是], [14, 15, 13, 13, 是], [14, 16, 12, 12, 是], [14, 17, 11, 11, 是], [14, 18, 10, 10, 是], [14, 19, 9, 9, 是], [14, 20, 8, 8, 是], [15, 0, 6, 6, 是], [15, 1, 5, 5, 是], [15, 2, 4, 4, 是], [15, 3, 3, 3, 是], [15, 4, 2, 2, 是], [15, 5, 1, 1, 是], [15, 6, 0, 0, 是], [15, 7, 20, 20, 是], [15, 8, 19, 19, 是], [15, 9, 18, 18, 是], [15, 10, 17, 17, 是], [15, 11, 16, 16, 是], [15, 12, 15, 15, 是], [15, 13, 14, 14, 是], [15, 14, 13, 13, 是], [15, 15, 12, 12, 是], [15, 16, 11, 11, 是], [15, 17, 10, 10, 是], [15, 18, 9, 9, 是], [15, 19, 8, 8, 是], [15, 20, 7, 7, 是], [16, 0, 5, 5, 是], [16, 1, 4, 4, 是], [16, 2, 3, 3, 是], [16, 3, 2, 2, 是], [16, 4, 1, 1, 是], [16, 5, 0, 0, 是], [16, 6, 20, 20, 是], [16, 7, 19, 19, 是], [16, 8, 18, 18, 是], [16, 9, 17, 17, 是], [16, 10, 16, 16, 是], [16, 11, 15, 15, 是], [16, 12, 14, 14, 是], [16, 13, 13, 13, 是], [16, 14, 12, 12, 是], [16, 15, 11, 11, 是], [16, 16, 10, 10, 是], [16, 17, 9, 9, 是], [16, 18, 8, 8, 是], [16, 19, 7, 7, 是], [16, 20, 6, 6, 是], [17, 0, 4, 4, 是], [17, 1, 3, 3, 是], [17, 2, 2, 2, 是], [17, 3, 1, 1, 是], [17, 4, 0, 0, 是], [17, 5, 20, 20, 是], [17, 6, 19, 19, 是], [17, 7, 18, 18, 是], [17, 8, 17, 17, 是], [17, 9, 16, 16, 是], [17, 10, 15, 15, 是], [17, 11, 14, 14, 是], [17, 12, 13, 13, 是], [17, 13, 12, 12, 是], [17, 14, 11, 11, 是], [17, 15, 10, 10, 是], [17, 16, 9, 9, 是], [17, 17, 8, 8, 是], [17, 18, 7, 7, 是], [17, 19, 6, 6, 是], [17, 20, 5, 5, 是], [18, 0, 3, 3, 是], [18, 1, 2, 2, 是], [18, 2, 1, 1, 是], [18, 3, 0, 0, 是], [18, 4, 20, 20, 是], [18, 5, 19, 19, 是], [18, 6, 18, 18, 是], [18, 7, 17, 17, 是], [18, 8, 16, 16, 是], [18, 9, 15, 15, 是], [18, 10, 14, 14, 是], [18, 11, 13, 13, 是], [18, 12, 12, 12, 是], [18, 13, 11, 11, 是], [18, 14, 10, 10, 是], [18, 15, 9, 9, 是], [18, 16, 8, 8, 是], [18, 17, 7, 7, 是], [18, 18, 6, 6, 是], [18, 19, 5, 5, 是], [18, 20, 4, 4, 是], [19, 0, 2, 2, 是], [19, 1, 1, 1, 是], [19, 2, 0, 0, 是], [19, 3, 20, 20, 是], [19, 4, 19, 19, 是], [19, 5, 18, 18, 是], [19, 6, 17, 17, 是], [19, 7, 16, 16, 是], [19, 8, 15, 15, 是], [19, 9, 14, 14, 是], [19, 10, 13, 13, 是], [19, 11, 12, 12, 是], [19, 12, 11, 11, 是], [19, 13, 10, 10, 是], [19, 14, 9, 9, 是], [19, 15, 8, 8, 是], [19, 16, 7, 7, 是], [19, 17, 6, 6, 是], [19, 18, 5, 5, 是], [19, 19, 4, 4, 是], [19, 20, 3, 3, 是], [20, 0, 1, 1, 是], [20, 1, 0, 0, 是], [20, 2, 20, 20, 是], [20, 3, 19, 19, 是], [20, 4, 18, 18, 是], [20, 5, 17, 17, 是], [20, 6, 16, 16, 是], [20, 7, 15, 15, 是], [20, 8, 14, 14, 是], [20, 9, 13, 13, 是], [20, 10, 12, 12, 是], [20, 11, 11, 11, 是], [20, 12, 10, 10, 是], [20, 13, 9, 9, 是], [20, 14, 8, 8, 是], [20, 15, 7, 7, 是], [20, 16, 6, 6, 是], [20, 17, 5, 5, 是], [20, 18, 4, 4, 是], [20, 19, 3, 3, 是], [20, 20, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(21, 21, [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 20, 20, 是], [0, 2, 19, 19, 是], [0, 3, 18, 18, 是], [0, 4, 17, 17, 是], [0, 5, 16, 16, 是], [0, 6, 15, 15, 是], [0, 7, 14, 14, 是], [0, 8, 13, 13, 是], [0, 9, 12, 12, 是], [0, 10, 11, 11, 是], [0, 11, 10, 10, 是], [0, 12, 9, 9, 是], [0, 13, 8, 8, 是], [0, 14, 7, 7, 是], [0, 15, 6, 6, 是], [0, 16, 5, 5, 是], [0, 17, 4, 4, 是], [0, 18, 3, 3, 是], [0, 19, 2, 2, 是], [0, 20, 1, 1, 是], [1, 0, 20, 20, 是], [1, 1, 19, 19, 是], [1, 2, 18, 18, 是], [1, 3, 17, 17, 是], [1, 4, 16, 16, 是], [1, 5, 15, 15, 是], [1, 6, 14, 14, 是], [1, 7, 13, 13, 是], [1, 8, 12, 12, 是], [1, 9, 11, 11, 是], [1, 10, 10, 10, 是], [1, 11, 9, 9, 是], [1, 12, 8, 8, 是], [1, 13, 7, 7, 是], [1, 14, 6, 6, 是], [1, 15, 5, 5, 是], [1, 16, 4, 4, 是], [1, 17, 3, 3, 是], [1, 18, 2, 2, 是], [1, 19, 1, 1, 是], [1, 20, 0, 0, 是], [2, 0, 19, 19, 是], [2, 1, 18, 18, 是], [2, 2, 17, 17, 是], [2, 3, 16, 16, 是], [2, 4, 15, 15, 是], [2, 5, 14, 14, 是], [2, 6, 13, 13, 是], [2, 7, 12, 12, 是], [2, 8, 11, 11, 是], [2, 9, 10, 10, 是], [2, 10, 9, 9, 是], [2, 11, 8, 8, 是], [2, 12, 7, 7, 是], [2, 13, 6, 6, 是], [2, 14, 5, 5, 是], [2, 15, 4, 4, 是], [2, 16, 3, 3, 是], [2, 17, 2, 2, 是], [2, 18, 1, 1, 是], [2, 19, 0, 0, 是], [2, 20, 20, 20, 是], [3, 0, 18, 18, 是], [3, 1, 17, 17, 是], [3, 2, 16, 16, 是], [3, 3, 15, 15, 是], [3, 4, 14, 14, 是], [3, 5, 13, 13, 是], [3, 6, 12, 12, 是], [3, 7, 11, 11, 是], [3, 8, 10, 10, 是], [3, 9, 9, 9, 是], [3, 10, 8, 8, 是], [3, 11, 7, 7, 是], [3, 12, 6, 6, 是], [3, 13, 5, 5, 是], [3, 14, 4, 4, 是], [3, 15, 3, 3, 是], [3, 16, 2, 2, 是], [3, 17, 1, 1, 是], [3, 18, 0, 0, 是], [3, 19, 20, 20, 是], [3, 20, 19, 19, 是], [4, 0, 17, 17, 是], [4, 1, 16, 16, 是], [4, 2, 15, 15, 是], [4, 3, 14, 14, 是], [4, 4, 13, 13, 是], [4, 5, 12, 12, 是], [4, 6, 11, 11, 是], [4, 7, 10, 10, 是], [4, 8, 9, 9, 是], [4, 9, 8, 8, 是], [4, 10, 7, 7, 是], [4, 11, 6, 6, 是], [4, 12, 5, 5, 是], [4, 13, 4, 4, 是], [4, 14, 3, 3, 是], [4, 15, 2, 2, 是], [4, 16, 1, 1, 是], [4, 17, 0, 0, 是], [4, 18, 20, 20, 是], [4, 19, 19, 19, 是], [4, 20, 18, 18, 是], [5, 0, 16, 16, 是], [5, 1, 15, 15, 是], [5, 2, 14, 14, 是], [5, 3, 13, 13, 是], [5, 4, 12, 12, 是], [5, 5, 11, 11, 是], [5, 6, 10, 10, 是], [5, 7, 9, 9, 是], [5, 8, 8, 8, 是], [5, 9, 7, 7, 是], [5, 10, 6, 6, 是], [5, 11, 5, 5, 是], [5, 12, 4, 4, 是], [5, 13, 3, 3, 是], [5, 14, 2, 2, 是], [5, 15, 1, 1, 是], [5, 16, 0, 0, 是], [5, 17, 20, 20, 是], [5, 18, 19, 19, 是], [5, 19, 18, 18, 是], [5, 20, 17, 17, 是], [6, 0, 15, 15, 是], [6, 1, 14, 14, 是], [6, 2, 13, 13, 是], [6, 3, 12, 12, 是], [6, 4, 11, 11, 是], [6, 5, 10, 10, 是], [6, 6, 9, 9, 是], [6, 7, 8, 8, 是], [6, 8, 7, 7, 是], [6, 9, 6, 6, 是], [6, 10, 5, 5, 是], [6, 11, 4, 4, 是], [6, 12, 3, 3, 是], [6, 13, 2, 2, 是], [6, 14, 1, 1, 是], [6, 15, 0, 0, 是], [6, 16, 20, 20, 是], [6, 17, 19, 19, 是], [6, 18, 18, 18, 是], [6, 19, 17, 17, 是], [6, 20, 16, 16, 是], [7, 0, 14, 14, 是], [7, 1, 13, 13, 是], [7, 2, 12, 12, 是], [7, 3, 11, 11, 是], [7, 4, 10, 10, 是], [7, 5, 9, 9, 是], [7, 6, 8, 8, 是], [7, 7, 7, 7, 是], [7, 8, 6, 6, 是], [7, 9, 5, 5, 是], [7, 10, 4, 4, 是], [7, 11, 3, 3, 是], [7, 12, 2, 2, 是], [7, 13, 1, 1, 是], [7, 14, 0, 0, 是], [7, 15, 20, 20, 是], [7, 16, 19, 19, 是], [7, 17, 18, 18, 是], [7, 18, 17, 17, 是], [7, 19, 16, 16, 是], [7, 20, 15, 15, 是], [8, 0, 13, 13, 是], [8, 1, 12, 12, 是], [8, 2, 11, 11, 是], [8, 3, 10, 10, 是], [8, 4, 9, 9, 是], [8, 5, 8, 8, 是], [8, 6, 7, 7, 是], [8, 7, 6, 6, 是], [8, 8, 5, 5, 是], [8, 9, 4, 4, 是], [8, 10, 3, 3, 是], [8, 11, 2, 2, 是], [8, 12, 1, 1, 是], [8, 13, 0, 0, 是], [8, 14, 20, 20, 是], [8, 15, 19, 19, 是], [8, 16, 18, 18, 是], [8, 17, 17, 17, 是], [8, 18, 16, 16, 是], [8, 19, 15, 15, 是], [8, 20, 14, 14, 是], [9, 0, 12, 12, 是], [9, 1, 11, 11, 是], [9, 2, 10, 10, 是], [9, 3, 9, 9, 是], [9, 4, 8, 8, 是], [9, 5, 7, 7, 是], [9, 6, 6, 6, 是], [9, 7, 5, 5, 是], [9, 8, 4, 4, 是], [9, 9, 3, 3, 是], [9, 10, 2, 2, 是], [9, 11, 1, 1, 是], [9, 12, 0, 0, 是], [9, 13, 20, 20, 是], [9, 14, 19, 19, 是], [9, 15, 18, 18, 是], [9, 16, 17, 17, 是], [9, 17, 16, 16, 是], [9, 18, 15, 15, 是], [9, 19, 14, 14, 是], [9, 20, 13, 13, 是], [10, 0, 11, 11, 是], [10, 1, 10, 10, 是], [10, 2, 9, 9, 是], [10, 3, 8, 8, 是], [10, 4, 7, 7, 是], [10, 5, 6, 6, 是], [10, 6, 5, 5, 是], [10, 7, 4, 4, 是], [10, 8, 3, 3, 是], [10, 9, 2, 2, 是], [10, 10, 1, 1, 是], [10, 11, 0, 0, 是], [10, 12, 20, 20, 是], [10, 13, 19, 19, 是], [10, 14, 18, 18, 是], [10, 15, 17, 17, 是], [10, 16, 16, 16, 是], [10, 17, 15, 15, 是], [10, 18, 14, 14, 是], [10, 19, 13, 13, 是], [10, 20, 12, 12, 是], [11, 0, 10, 10, 是], [11, 1, 9, 9, 是], [11, 2, 8, 8, 是], [11, 3, 7, 7, 是], [11, 4, 6, 6, 是], [11, 5, 5, 5, 是], [11, 6, 4, 4, 是], [11, 7, 3, 3, 是], [11, 8, 2, 2, 是], [11, 9, 1, 1, 是], [11, 10, 0, 0, 是], [11, 11, 20, 20, 是], [11, 12, 19, 19, 是], [11, 13, 18, 18, 是], [11, 14, 17, 17, 是], [11, 15, 16, 16, 是], [11, 16, 15, 15, 是], [11, 17, 14, 14, 是], [11, 18, 13, 13, 是], [11, 19, 12, 12, 是], [11, 20, 11, 11, 是], [12, 0, 9, 9, 是], [12, 1, 8, 8, 是], [12, 2, 7, 7, 是], [12, 3, 6, 6, 是], [12, 4, 5, 5, 是], [12, 5, 4, 4, 是], [12, 6, 3, 3, 是], [12, 7, 2, 2, 是], [12, 8, 1, 1, 是], [12, 9, 0, 0, 是], [12, 10, 20, 20, 是], [12, 11, 19, 19, 是], [12, 12, 18, 18, 是], [12, 13, 17, 17, 是], [12, 14, 16, 16, 是], [12, 15, 15, 15, 是], [12, 16, 14, 14, 是], [12, 17, 13, 13, 是], [12, 18, 12, 12, 是], [12, 19, 11, 11, 是], [12, 20, 10, 10, 是], [13, 0, 8, 8, 是], [13, 1, 7, 7, 是], [13, 2, 6, 6, 是], [13, 3, 5, 5, 是], [13, 4, 4, 4, 是], [13, 5, 3, 3, 是], [13, 6, 2, 2, 是], [13, 7, 1, 1, 是], [13, 8, 0, 0, 是], [13, 9, 20, 20, 是], [13, 10, 19, 19, 是], [13, 11, 18, 18, 是], [13, 12, 17, 17, 是], [13, 13, 16, 16, 是], [13, 14, 15, 15, 是], [13, 15, 14, 14, 是], [13, 16, 13, 13, 是], [13, 17, 12, 12, 是], [13, 18, 11, 11, 是], [13, 19, 10, 10, 是], [13, 20, 9, 9, 是], [14, 0, 7, 7, 是], [14, 1, 6, 6, 是], [14, 2, 5, 5, 是], [14, 3, 4, 4, 是], [14, 4, 3, 3, 是], [14, 5, 2, 2, 是], [14, 6, 1, 1, 是], [14, 7, 0, 0, 是], [14, 8, 20, 20, 是], [14, 9, 19, 19, 是], [14, 10, 18, 18, 是], [14, 11, 17, 17, 是], [14, 12, 16, 16, 是], [14, 13, 15, 15, 是], [14, 14, 14, 14, 是], [14, 15, 13, 13, 是], [14, 16, 12, 12, 是], [14, 17, 11, 11, 是], [14, 18, 10, 10, 是], [14, 19, 9, 9, 是], [14, 20, 8, 8, 是], [15, 0, 6, 6, 是], [15, 1, 5, 5, 是], [15, 2, 4, 4, 是], [15, 3, 3, 3, 是], [15, 4, 2, 2, 是], [15, 5, 1, 1, 是], [15, 6, 0, 0, 是], [15, 7, 20, 20, 是], [15, 8, 19, 19, 是], [15, 9, 18, 18, 是], [15, 10, 17, 17, 是], [15, 11, 16, 16, 是], [15, 12, 15, 15, 是], [15, 13, 14, 14, 是], [15, 14, 13, 13, 是], [15, 15, 12, 12, 是], [15, 16, 11, 11, 是], [15, 17, 10, 10, 是], [15, 18, 9, 9, 是], [15, 19, 8, 8, 是], [15, 20, 7, 7, 是], [16, 0, 5, 5, 是], [16, 1, 4, 4, 是], [16, 2, 3, 3, 是], [16, 3, 2, 2, 是], [16, 4, 1, 1, 是], [16, 5, 0, 0, 是], [16, 6, 20, 20, 是], [16, 7, 19, 19, 是], [16, 8, 18, 18, 是], [16, 9, 17, 17, 是], [16, 10, 16, 16, 是], [16, 11, 15, 15, 是], [16, 12, 14, 14, 是], [16, 13, 13, 13, 是], [16, 14, 12, 12, 是], [16, 15, 11, 11, 是], [16, 16, 10, 10, 是], [16, 17, 9, 9, 是], [16, 18, 8, 8, 是], [16, 19, 7, 7, 是], [16, 20, 6, 6, 是], [17, 0, 4, 4, 是], [17, 1, 3, 3, 是], [17, 2, 2, 2, 是], [17, 3, 1, 1, 是], [17, 4, 0, 0, 是], [17, 5, 20, 20, 是], [17, 6, 19, 19, 是], [17, 7, 18, 18, 是], [17, 8, 17, 17, 是], [17, 9, 16, 16, 是], [17, 10, 15, 15, 是], [17, 11, 14, 14, 是], [17, 12, 13, 13, 是], [17, 13, 12, 12, 是], [17, 14, 11, 11, 是], [17, 15, 10, 10, 是], [17, 16, 9, 9, 是], [17, 17, 8, 8, 是], [17, 18, 7, 7, 是], [17, 19, 6, 6, 是], [17, 20, 5, 5, 是], [18, 0, 3, 3, 是], [18, 1, 2, 2, 是], [18, 2, 1, 1, 是], [18, 3, 0, 0, 是], [18, 4, 20, 20, 是], [18, 5, 19, 19, 是], [18, 6, 18, 18, 是], [18, 7, 17, 17, 是], [18, 8, 16, 16, 是], [18, 9, 15, 15, 是], [18, 10, 14, 14, 是], [18, 11, 13, 13, 是], [18, 12, 12, 12, 是], [18, 13, 11, 11, 是], [18, 14, 10, 10, 是], [18, 15, 9, 9, 是], [18, 16, 8, 8, 是], [18, 17, 7, 7, 是], [18, 18, 6, 6, 是], [18, 19, 5, 5, 是], [18, 20, 4, 4, 是], [19, 0, 2, 2, 是], [19, 1, 1, 1, 是], [19, 2, 0, 0, 是], [19, 3, 20, 20, 是], [19, 4, 19, 19, 是], [19, 5, 18, 18, 是], [19, 6, 17, 17, 是], [19, 7, 16, 16, 是], [19, 8, 15, 15, 是], [19, 9, 14, 14, 是], [19, 10, 13, 13, 是], [19, 11, 12, 12, 是], [19, 12, 11, 11, 是], [19, 13, 10, 10, 是], [19, 14, 9, 9, 是], [19, 15, 8, 8, 是], [19, 16, 7, 7, 是], [19, 17, 6, 6, 是], [19, 18, 5, 5, 是], [19, 19, 4, 4, 是], [19, 20, 3, 3, 是], [20, 0, 1, 1, 是], [20, 1, 0, 0, 是], [20, 2, 20, 20, 是], [20, 3, 19, 19, 是], [20, 4, 18, 18, 是], [20, 5, 17, 17, 是], [20, 6, 16, 16, 是], [20, 7, 15, 15, 是], [20, 8, 14, 14, 是], [20, 9, 13, 13, 是], [20, 10, 12, 12, 是], [20, 11, 11, 11, 是], [20, 12, 10, 10, 是], [20, 13, 9, 9, 是], [20, 14, 8, 8, 是], [20, 15, 7, 7, 是], [20, 16, 6, 6, 是], [20, 17, 5, 5, 是], [20, 18, 4, 4, 是], [20, 19, 3, 3, 是], [20, 20, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141019,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],"target_identity":0,"mapping":[0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0020","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"mapping\":[0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 22, [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 21, 21, 是], [0, 2, 20, 20, 是], [0, 3, 19, 19, 是], [0, 4, 18, 18, 是], [0, 5, 17, 17, 是], [0, 6, 16, 16, 是], [0, 7, 15, 15, 是], [0, 8, 14, 14, 是], [0, 9, 13, 13, 是], [0, 10, 12, 12, 是], [0, 11, 11, 11, 是], [0, 12, 10, 10, 是], [0, 13, 9, 9, 是], [0, 14, 8, 8, 是], [0, 15, 7, 7, 是], [0, 16, 6, 6, 是], [0, 17, 5, 5, 是], [0, 18, 4, 4, 是], [0, 19, 3, 3, 是], [0, 20, 2, 2, 是], [0, 21, 1, 1, 是], [1, 0, 21, 21, 是], [1, 1, 20, 20, 是], [1, 2, 19, 19, 是], [1, 3, 18, 18, 是], [1, 4, 17, 17, 是], [1, 5, 16, 16, 是], [1, 6, 15, 15, 是], [1, 7, 14, 14, 是], [1, 8, 13, 13, 是], [1, 9, 12, 12, 是], [1, 10, 11, 11, 是], [1, 11, 10, 10, 是], [1, 12, 9, 9, 是], [1, 13, 8, 8, 是], [1, 14, 7, 7, 是], [1, 15, 6, 6, 是], [1, 16, 5, 5, 是], [1, 17, 4, 4, 是], [1, 18, 3, 3, 是], [1, 19, 2, 2, 是], [1, 20, 1, 1, 是], [1, 21, 0, 0, 是], [2, 0, 20, 20, 是], [2, 1, 19, 19, 是], [2, 2, 18, 18, 是], [2, 3, 17, 17, 是], [2, 4, 16, 16, 是], [2, 5, 15, 15, 是], [2, 6, 14, 14, 是], [2, 7, 13, 13, 是], [2, 8, 12, 12, 是], [2, 9, 11, 11, 是], [2, 10, 10, 10, 是], [2, 11, 9, 9, 是], [2, 12, 8, 8, 是], [2, 13, 7, 7, 是], [2, 14, 6, 6, 是], [2, 15, 5, 5, 是], [2, 16, 4, 4, 是], [2, 17, 3, 3, 是], [2, 18, 2, 2, 是], [2, 19, 1, 1, 是], [2, 20, 0, 0, 是], [2, 21, 21, 21, 是], [3, 0, 19, 19, 是], [3, 1, 18, 18, 是], [3, 2, 17, 17, 是], [3, 3, 16, 16, 是], [3, 4, 15, 15, 是], [3, 5, 14, 14, 是], [3, 6, 13, 13, 是], [3, 7, 12, 12, 是], [3, 8, 11, 11, 是], [3, 9, 10, 10, 是], [3, 10, 9, 9, 是], [3, 11, 8, 8, 是], [3, 12, 7, 7, 是], [3, 13, 6, 6, 是], [3, 14, 5, 5, 是], [3, 15, 4, 4, 是], [3, 16, 3, 3, 是], [3, 17, 2, 2, 是], [3, 18, 1, 1, 是], [3, 19, 0, 0, 是], [3, 20, 21, 21, 是], [3, 21, 20, 20, 是], [4, 0, 18, 18, 是], [4, 1, 17, 17, 是], [4, 2, 16, 16, 是], [4, 3, 15, 15, 是], [4, 4, 14, 14, 是], [4, 5, 13, 13, 是], [4, 6, 12, 12, 是], [4, 7, 11, 11, 是], [4, 8, 10, 10, 是], [4, 9, 9, 9, 是], [4, 10, 8, 8, 是], [4, 11, 7, 7, 是], [4, 12, 6, 6, 是], [4, 13, 5, 5, 是], [4, 14, 4, 4, 是], [4, 15, 3, 3, 是], [4, 16, 2, 2, 是], [4, 17, 1, 1, 是], [4, 18, 0, 0, 是], [4, 19, 21, 21, 是], [4, 20, 20, 20, 是], [4, 21, 19, 19, 是], [5, 0, 17, 17, 是], [5, 1, 16, 16, 是], [5, 2, 15, 15, 是], [5, 3, 14, 14, 是], [5, 4, 13, 13, 是], [5, 5, 12, 12, 是], [5, 6, 11, 11, 是], [5, 7, 10, 10, 是], [5, 8, 9, 9, 是], [5, 9, 8, 8, 是], [5, 10, 7, 7, 是], [5, 11, 6, 6, 是], [5, 12, 5, 5, 是], [5, 13, 4, 4, 是], [5, 14, 3, 3, 是], [5, 15, 2, 2, 是], [5, 16, 1, 1, 是], [5, 17, 0, 0, 是], [5, 18, 21, 21, 是], [5, 19, 20, 20, 是], [5, 20, 19, 19, 是], [5, 21, 18, 18, 是], [6, 0, 16, 16, 是], [6, 1, 15, 15, 是], [6, 2, 14, 14, 是], [6, 3, 13, 13, 是], [6, 4, 12, 12, 是], [6, 5, 11, 11, 是], [6, 6, 10, 10, 是], [6, 7, 9, 9, 是], [6, 8, 8, 8, 是], [6, 9, 7, 7, 是], [6, 10, 6, 6, 是], [6, 11, 5, 5, 是], [6, 12, 4, 4, 是], [6, 13, 3, 3, 是], [6, 14, 2, 2, 是], [6, 15, 1, 1, 是], [6, 16, 0, 0, 是], [6, 17, 21, 21, 是], [6, 18, 20, 20, 是], [6, 19, 19, 19, 是], [6, 20, 18, 18, 是], [6, 21, 17, 17, 是], [7, 0, 15, 15, 是], [7, 1, 14, 14, 是], [7, 2, 13, 13, 是], [7, 3, 12, 12, 是], [7, 4, 11, 11, 是], [7, 5, 10, 10, 是], [7, 6, 9, 9, 是], [7, 7, 8, 8, 是], [7, 8, 7, 7, 是], [7, 9, 6, 6, 是], [7, 10, 5, 5, 是], [7, 11, 4, 4, 是], [7, 12, 3, 3, 是], [7, 13, 2, 2, 是], [7, 14, 1, 1, 是], [7, 15, 0, 0, 是], [7, 16, 21, 21, 是], [7, 17, 20, 20, 是], [7, 18, 19, 19, 是], [7, 19, 18, 18, 是], [7, 20, 17, 17, 是], [7, 21, 16, 16, 是], [8, 0, 14, 14, 是], [8, 1, 13, 13, 是], [8, 2, 12, 12, 是], [8, 3, 11, 11, 是], [8, 4, 10, 10, 是], [8, 5, 9, 9, 是], [8, 6, 8, 8, 是], [8, 7, 7, 7, 是], [8, 8, 6, 6, 是], [8, 9, 5, 5, 是], [8, 10, 4, 4, 是], [8, 11, 3, 3, 是], [8, 12, 2, 2, 是], [8, 13, 1, 1, 是], [8, 14, 0, 0, 是], [8, 15, 21, 21, 是], [8, 16, 20, 20, 是], [8, 17, 19, 19, 是], [8, 18, 18, 18, 是], [8, 19, 17, 17, 是], [8, 20, 16, 16, 是], [8, 21, 15, 15, 是], [9, 0, 13, 13, 是], [9, 1, 12, 12, 是], [9, 2, 11, 11, 是], [9, 3, 10, 10, 是], [9, 4, 9, 9, 是], [9, 5, 8, 8, 是], [9, 6, 7, 7, 是], [9, 7, 6, 6, 是], [9, 8, 5, 5, 是], [9, 9, 4, 4, 是], [9, 10, 3, 3, 是], [9, 11, 2, 2, 是], [9, 12, 1, 1, 是], [9, 13, 0, 0, 是], [9, 14, 21, 21, 是], [9, 15, 20, 20, 是], [9, 16, 19, 19, 是], [9, 17, 18, 18, 是], [9, 18, 17, 17, 是], [9, 19, 16, 16, 是], [9, 20, 15, 15, 是], [9, 21, 14, 14, 是], [10, 0, 12, 12, 是], [10, 1, 11, 11, 是], [10, 2, 10, 10, 是], [10, 3, 9, 9, 是], [10, 4, 8, 8, 是], [10, 5, 7, 7, 是], [10, 6, 6, 6, 是], [10, 7, 5, 5, 是], [10, 8, 4, 4, 是], [10, 9, 3, 3, 是], [10, 10, 2, 2, 是], [10, 11, 1, 1, 是], [10, 12, 0, 0, 是], [10, 13, 21, 21, 是], [10, 14, 20, 20, 是], [10, 15, 19, 19, 是], [10, 16, 18, 18, 是], [10, 17, 17, 17, 是], [10, 18, 16, 16, 是], [10, 19, 15, 15, 是], [10, 20, 14, 14, 是], [10, 21, 13, 13, 是], [11, 0, 11, 11, 是], [11, 1, 10, 10, 是], [11, 2, 9, 9, 是], [11, 3, 8, 8, 是], [11, 4, 7, 7, 是], [11, 5, 6, 6, 是], [11, 6, 5, 5, 是], [11, 7, 4, 4, 是], [11, 8, 3, 3, 是], [11, 9, 2, 2, 是], [11, 10, 1, 1, 是], [11, 11, 0, 0, 是], [11, 12, 21, 21, 是], [11, 13, 20, 20, 是], [11, 14, 19, 19, 是], [11, 15, 18, 18, 是], [11, 16, 17, 17, 是], [11, 17, 16, 16, 是], [11, 18, 15, 15, 是], [11, 19, 14, 14, 是], [11, 20, 13, 13, 是], [11, 21, 12, 12, 是], [12, 0, 10, 10, 是], [12, 1, 9, 9, 是], [12, 2, 8, 8, 是], [12, 3, 7, 7, 是], [12, 4, 6, 6, 是], [12, 5, 5, 5, 是], [12, 6, 4, 4, 是], [12, 7, 3, 3, 是], [12, 8, 2, 2, 是], [12, 9, 1, 1, 是], [12, 10, 0, 0, 是], [12, 11, 21, 21, 是], [12, 12, 20, 20, 是], [12, 13, 19, 19, 是], [12, 14, 18, 18, 是], [12, 15, 17, 17, 是], [12, 16, 16, 16, 是], [12, 17, 15, 15, 是], [12, 18, 14, 14, 是], [12, 19, 13, 13, 是], [12, 20, 12, 12, 是], [12, 21, 11, 11, 是], [13, 0, 9, 9, 是], [13, 1, 8, 8, 是], [13, 2, 7, 7, 是], [13, 3, 6, 6, 是], [13, 4, 5, 5, 是], [13, 5, 4, 4, 是], [13, 6, 3, 3, 是], [13, 7, 2, 2, 是], [13, 8, 1, 1, 是], [13, 9, 0, 0, 是], [13, 10, 21, 21, 是], [13, 11, 20, 20, 是], [13, 12, 19, 19, 是], [13, 13, 18, 18, 是], [13, 14, 17, 17, 是], [13, 15, 16, 16, 是], [13, 16, 15, 15, 是], [13, 17, 14, 14, 是], [13, 18, 13, 13, 是], [13, 19, 12, 12, 是], [13, 20, 11, 11, 是], [13, 21, 10, 10, 是], [14, 0, 8, 8, 是], [14, 1, 7, 7, 是], [14, 2, 6, 6, 是], [14, 3, 5, 5, 是], [14, 4, 4, 4, 是], [14, 5, 3, 3, 是], [14, 6, 2, 2, 是], [14, 7, 1, 1, 是], [14, 8, 0, 0, 是], [14, 9, 21, 21, 是], [14, 10, 20, 20, 是], [14, 11, 19, 19, 是], [14, 12, 18, 18, 是], [14, 13, 17, 17, 是], [14, 14, 16, 16, 是], [14, 15, 15, 15, 是], [14, 16, 14, 14, 是], [14, 17, 13, 13, 是], [14, 18, 12, 12, 是], [14, 19, 11, 11, 是], [14, 20, 10, 10, 是], [14, 21, 9, 9, 是], [15, 0, 7, 7, 是], [15, 1, 6, 6, 是], [15, 2, 5, 5, 是], [15, 3, 4, 4, 是], [15, 4, 3, 3, 是], [15, 5, 2, 2, 是], [15, 6, 1, 1, 是], [15, 7, 0, 0, 是], [15, 8, 21, 21, 是], [15, 9, 20, 20, 是], [15, 10, 19, 19, 是], [15, 11, 18, 18, 是], [15, 12, 17, 17, 是], [15, 13, 16, 16, 是], [15, 14, 15, 15, 是], [15, 15, 14, 14, 是], [15, 16, 13, 13, 是], [15, 17, 12, 12, 是], [15, 18, 11, 11, 是], [15, 19, 10, 10, 是], [15, 20, 9, 9, 是], [15, 21, 8, 8, 是], [16, 0, 6, 6, 是], [16, 1, 5, 5, 是], [16, 2, 4, 4, 是], [16, 3, 3, 3, 是], [16, 4, 2, 2, 是], [16, 5, 1, 1, 是], [16, 6, 0, 0, 是], [16, 7, 21, 21, 是], [16, 8, 20, 20, 是], [16, 9, 19, 19, 是], [16, 10, 18, 18, 是], [16, 11, 17, 17, 是], [16, 12, 16, 16, 是], [16, 13, 15, 15, 是], [16, 14, 14, 14, 是], [16, 15, 13, 13, 是], [16, 16, 12, 12, 是], [16, 17, 11, 11, 是], [16, 18, 10, 10, 是], [16, 19, 9, 9, 是], [16, 20, 8, 8, 是], [16, 21, 7, 7, 是], [17, 0, 5, 5, 是], [17, 1, 4, 4, 是], [17, 2, 3, 3, 是], [17, 3, 2, 2, 是], [17, 4, 1, 1, 是], [17, 5, 0, 0, 是], [17, 6, 21, 21, 是], [17, 7, 20, 20, 是], [17, 8, 19, 19, 是], [17, 9, 18, 18, 是], [17, 10, 17, 17, 是], [17, 11, 16, 16, 是], [17, 12, 15, 15, 是], [17, 13, 14, 14, 是], [17, 14, 13, 13, 是], [17, 15, 12, 12, 是], [17, 16, 11, 11, 是], [17, 17, 10, 10, 是], [17, 18, 9, 9, 是], [17, 19, 8, 8, 是], [17, 20, 7, 7, 是], [17, 21, 6, 6, 是], [18, 0, 4, 4, 是], [18, 1, 3, 3, 是], [18, 2, 2, 2, 是], [18, 3, 1, 1, 是], [18, 4, 0, 0, 是], [18, 5, 21, 21, 是], [18, 6, 20, 20, 是], [18, 7, 19, 19, 是], [18, 8, 18, 18, 是], [18, 9, 17, 17, 是], [18, 10, 16, 16, 是], [18, 11, 15, 15, 是], [18, 12, 14, 14, 是], [18, 13, 13, 13, 是], [18, 14, 12, 12, 是], [18, 15, 11, 11, 是], [18, 16, 10, 10, 是], [18, 17, 9, 9, 是], [18, 18, 8, 8, 是], [18, 19, 7, 7, 是], [18, 20, 6, 6, 是], [18, 21, 5, 5, 是], [19, 0, 3, 3, 是], [19, 1, 2, 2, 是], [19, 2, 1, 1, 是], [19, 3, 0, 0, 是], [19, 4, 21, 21, 是], [19, 5, 20, 20, 是], [19, 6, 19, 19, 是], [19, 7, 18, 18, 是], [19, 8, 17, 17, 是], [19, 9, 16, 16, 是], [19, 10, 15, 15, 是], [19, 11, 14, 14, 是], [19, 12, 13, 13, 是], [19, 13, 12, 12, 是], [19, 14, 11, 11, 是], [19, 15, 10, 10, 是], [19, 16, 9, 9, 是], [19, 17, 8, 8, 是], [19, 18, 7, 7, 是], [19, 19, 6, 6, 是], [19, 20, 5, 5, 是], [19, 21, 4, 4, 是], [20, 0, 2, 2, 是], [20, 1, 1, 1, 是], [20, 2, 0, 0, 是], [20, 3, 21, 21, 是], [20, 4, 20, 20, 是], [20, 5, 19, 19, 是], [20, 6, 18, 18, 是], [20, 7, 17, 17, 是], [20, 8, 16, 16, 是], [20, 9, 15, 15, 是], [20, 10, 14, 14, 是], [20, 11, 13, 13, 是], [20, 12, 12, 12, 是], [20, 13, 11, 11, 是], [20, 14, 10, 10, 是], [20, 15, 9, 9, 是], [20, 16, 8, 8, 是], [20, 17, 7, 7, 是], [20, 18, 6, 6, 是], [20, 19, 5, 5, 是], [20, 20, 4, 4, 是], [20, 21, 3, 3, 是], [21, 0, 1, 1, 是], [21, 1, 0, 0, 是], [21, 2, 21, 21, 是], [21, 3, 20, 20, 是], [21, 4, 19, 19, 是], [21, 5, 18, 18, 是], [21, 6, 17, 17, 是], [21, 7, 16, 16, 是], [21, 8, 15, 15, 是], [21, 9, 14, 14, 是], [21, 10, 13, 13, 是], [21, 11, 12, 12, 是], [21, 12, 11, 11, 是], [21, 13, 10, 10, 是], [21, 14, 9, 9, 是], [21, 15, 8, 8, 是], [21, 16, 7, 7, 是], [21, 17, 6, 6, 是], [21, 18, 5, 5, 是], [21, 19, 4, 4, 是], [21, 20, 3, 3, 是], [21, 21, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(22, 22, [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 21, 21, 是], [0, 2, 20, 20, 是], [0, 3, 19, 19, 是], [0, 4, 18, 18, 是], [0, 5, 17, 17, 是], [0, 6, 16, 16, 是], [0, 7, 15, 15, 是], [0, 8, 14, 14, 是], [0, 9, 13, 13, 是], [0, 10, 12, 12, 是], [0, 11, 11, 11, 是], [0, 12, 10, 10, 是], [0, 13, 9, 9, 是], [0, 14, 8, 8, 是], [0, 15, 7, 7, 是], [0, 16, 6, 6, 是], [0, 17, 5, 5, 是], [0, 18, 4, 4, 是], [0, 19, 3, 3, 是], [0, 20, 2, 2, 是], [0, 21, 1, 1, 是], [1, 0, 21, 21, 是], [1, 1, 20, 20, 是], [1, 2, 19, 19, 是], [1, 3, 18, 18, 是], [1, 4, 17, 17, 是], [1, 5, 16, 16, 是], [1, 6, 15, 15, 是], [1, 7, 14, 14, 是], [1, 8, 13, 13, 是], [1, 9, 12, 12, 是], [1, 10, 11, 11, 是], [1, 11, 10, 10, 是], [1, 12, 9, 9, 是], [1, 13, 8, 8, 是], [1, 14, 7, 7, 是], [1, 15, 6, 6, 是], [1, 16, 5, 5, 是], [1, 17, 4, 4, 是], [1, 18, 3, 3, 是], [1, 19, 2, 2, 是], [1, 20, 1, 1, 是], [1, 21, 0, 0, 是], [2, 0, 20, 20, 是], [2, 1, 19, 19, 是], [2, 2, 18, 18, 是], [2, 3, 17, 17, 是], [2, 4, 16, 16, 是], [2, 5, 15, 15, 是], [2, 6, 14, 14, 是], [2, 7, 13, 13, 是], [2, 8, 12, 12, 是], [2, 9, 11, 11, 是], [2, 10, 10, 10, 是], [2, 11, 9, 9, 是], [2, 12, 8, 8, 是], [2, 13, 7, 7, 是], [2, 14, 6, 6, 是], [2, 15, 5, 5, 是], [2, 16, 4, 4, 是], [2, 17, 3, 3, 是], [2, 18, 2, 2, 是], [2, 19, 1, 1, 是], [2, 20, 0, 0, 是], [2, 21, 21, 21, 是], [3, 0, 19, 19, 是], [3, 1, 18, 18, 是], [3, 2, 17, 17, 是], [3, 3, 16, 16, 是], [3, 4, 15, 15, 是], [3, 5, 14, 14, 是], [3, 6, 13, 13, 是], [3, 7, 12, 12, 是], [3, 8, 11, 11, 是], [3, 9, 10, 10, 是], [3, 10, 9, 9, 是], [3, 11, 8, 8, 是], [3, 12, 7, 7, 是], [3, 13, 6, 6, 是], [3, 14, 5, 5, 是], [3, 15, 4, 4, 是], [3, 16, 3, 3, 是], [3, 17, 2, 2, 是], [3, 18, 1, 1, 是], [3, 19, 0, 0, 是], [3, 20, 21, 21, 是], [3, 21, 20, 20, 是], [4, 0, 18, 18, 是], [4, 1, 17, 17, 是], [4, 2, 16, 16, 是], [4, 3, 15, 15, 是], [4, 4, 14, 14, 是], [4, 5, 13, 13, 是], [4, 6, 12, 12, 是], [4, 7, 11, 11, 是], [4, 8, 10, 10, 是], [4, 9, 9, 9, 是], [4, 10, 8, 8, 是], [4, 11, 7, 7, 是], [4, 12, 6, 6, 是], [4, 13, 5, 5, 是], [4, 14, 4, 4, 是], [4, 15, 3, 3, 是], [4, 16, 2, 2, 是], [4, 17, 1, 1, 是], [4, 18, 0, 0, 是], [4, 19, 21, 21, 是], [4, 20, 20, 20, 是], [4, 21, 19, 19, 是], [5, 0, 17, 17, 是], [5, 1, 16, 16, 是], [5, 2, 15, 15, 是], [5, 3, 14, 14, 是], [5, 4, 13, 13, 是], [5, 5, 12, 12, 是], [5, 6, 11, 11, 是], [5, 7, 10, 10, 是], [5, 8, 9, 9, 是], [5, 9, 8, 8, 是], [5, 10, 7, 7, 是], [5, 11, 6, 6, 是], [5, 12, 5, 5, 是], [5, 13, 4, 4, 是], [5, 14, 3, 3, 是], [5, 15, 2, 2, 是], [5, 16, 1, 1, 是], [5, 17, 0, 0, 是], [5, 18, 21, 21, 是], [5, 19, 20, 20, 是], [5, 20, 19, 19, 是], [5, 21, 18, 18, 是], [6, 0, 16, 16, 是], [6, 1, 15, 15, 是], [6, 2, 14, 14, 是], [6, 3, 13, 13, 是], [6, 4, 12, 12, 是], [6, 5, 11, 11, 是], [6, 6, 10, 10, 是], [6, 7, 9, 9, 是], [6, 8, 8, 8, 是], [6, 9, 7, 7, 是], [6, 10, 6, 6, 是], [6, 11, 5, 5, 是], [6, 12, 4, 4, 是], [6, 13, 3, 3, 是], [6, 14, 2, 2, 是], [6, 15, 1, 1, 是], [6, 16, 0, 0, 是], [6, 17, 21, 21, 是], [6, 18, 20, 20, 是], [6, 19, 19, 19, 是], [6, 20, 18, 18, 是], [6, 21, 17, 17, 是], [7, 0, 15, 15, 是], [7, 1, 14, 14, 是], [7, 2, 13, 13, 是], [7, 3, 12, 12, 是], [7, 4, 11, 11, 是], [7, 5, 10, 10, 是], [7, 6, 9, 9, 是], [7, 7, 8, 8, 是], [7, 8, 7, 7, 是], [7, 9, 6, 6, 是], [7, 10, 5, 5, 是], [7, 11, 4, 4, 是], [7, 12, 3, 3, 是], [7, 13, 2, 2, 是], [7, 14, 1, 1, 是], [7, 15, 0, 0, 是], [7, 16, 21, 21, 是], [7, 17, 20, 20, 是], [7, 18, 19, 19, 是], [7, 19, 18, 18, 是], [7, 20, 17, 17, 是], [7, 21, 16, 16, 是], [8, 0, 14, 14, 是], [8, 1, 13, 13, 是], [8, 2, 12, 12, 是], [8, 3, 11, 11, 是], [8, 4, 10, 10, 是], [8, 5, 9, 9, 是], [8, 6, 8, 8, 是], [8, 7, 7, 7, 是], [8, 8, 6, 6, 是], [8, 9, 5, 5, 是], [8, 10, 4, 4, 是], [8, 11, 3, 3, 是], [8, 12, 2, 2, 是], [8, 13, 1, 1, 是], [8, 14, 0, 0, 是], [8, 15, 21, 21, 是], [8, 16, 20, 20, 是], [8, 17, 19, 19, 是], [8, 18, 18, 18, 是], [8, 19, 17, 17, 是], [8, 20, 16, 16, 是], [8, 21, 15, 15, 是], [9, 0, 13, 13, 是], [9, 1, 12, 12, 是], [9, 2, 11, 11, 是], [9, 3, 10, 10, 是], [9, 4, 9, 9, 是], [9, 5, 8, 8, 是], [9, 6, 7, 7, 是], [9, 7, 6, 6, 是], [9, 8, 5, 5, 是], [9, 9, 4, 4, 是], [9, 10, 3, 3, 是], [9, 11, 2, 2, 是], [9, 12, 1, 1, 是], [9, 13, 0, 0, 是], [9, 14, 21, 21, 是], [9, 15, 20, 20, 是], [9, 16, 19, 19, 是], [9, 17, 18, 18, 是], [9, 18, 17, 17, 是], [9, 19, 16, 16, 是], [9, 20, 15, 15, 是], [9, 21, 14, 14, 是], [10, 0, 12, 12, 是], [10, 1, 11, 11, 是], [10, 2, 10, 10, 是], [10, 3, 9, 9, 是], [10, 4, 8, 8, 是], [10, 5, 7, 7, 是], [10, 6, 6, 6, 是], [10, 7, 5, 5, 是], [10, 8, 4, 4, 是], [10, 9, 3, 3, 是], [10, 10, 2, 2, 是], [10, 11, 1, 1, 是], [10, 12, 0, 0, 是], [10, 13, 21, 21, 是], [10, 14, 20, 20, 是], [10, 15, 19, 19, 是], [10, 16, 18, 18, 是], [10, 17, 17, 17, 是], [10, 18, 16, 16, 是], [10, 19, 15, 15, 是], [10, 20, 14, 14, 是], [10, 21, 13, 13, 是], [11, 0, 11, 11, 是], [11, 1, 10, 10, 是], [11, 2, 9, 9, 是], [11, 3, 8, 8, 是], [11, 4, 7, 7, 是], [11, 5, 6, 6, 是], [11, 6, 5, 5, 是], [11, 7, 4, 4, 是], [11, 8, 3, 3, 是], [11, 9, 2, 2, 是], [11, 10, 1, 1, 是], [11, 11, 0, 0, 是], [11, 12, 21, 21, 是], [11, 13, 20, 20, 是], [11, 14, 19, 19, 是], [11, 15, 18, 18, 是], [11, 16, 17, 17, 是], [11, 17, 16, 16, 是], [11, 18, 15, 15, 是], [11, 19, 14, 14, 是], [11, 20, 13, 13, 是], [11, 21, 12, 12, 是], [12, 0, 10, 10, 是], [12, 1, 9, 9, 是], [12, 2, 8, 8, 是], [12, 3, 7, 7, 是], [12, 4, 6, 6, 是], [12, 5, 5, 5, 是], [12, 6, 4, 4, 是], [12, 7, 3, 3, 是], [12, 8, 2, 2, 是], [12, 9, 1, 1, 是], [12, 10, 0, 0, 是], [12, 11, 21, 21, 是], [12, 12, 20, 20, 是], [12, 13, 19, 19, 是], [12, 14, 18, 18, 是], [12, 15, 17, 17, 是], [12, 16, 16, 16, 是], [12, 17, 15, 15, 是], [12, 18, 14, 14, 是], [12, 19, 13, 13, 是], [12, 20, 12, 12, 是], [12, 21, 11, 11, 是], [13, 0, 9, 9, 是], [13, 1, 8, 8, 是], [13, 2, 7, 7, 是], [13, 3, 6, 6, 是], [13, 4, 5, 5, 是], [13, 5, 4, 4, 是], [13, 6, 3, 3, 是], [13, 7, 2, 2, 是], [13, 8, 1, 1, 是], [13, 9, 0, 0, 是], [13, 10, 21, 21, 是], [13, 11, 20, 20, 是], [13, 12, 19, 19, 是], [13, 13, 18, 18, 是], [13, 14, 17, 17, 是], [13, 15, 16, 16, 是], [13, 16, 15, 15, 是], [13, 17, 14, 14, 是], [13, 18, 13, 13, 是], [13, 19, 12, 12, 是], [13, 20, 11, 11, 是], [13, 21, 10, 10, 是], [14, 0, 8, 8, 是], [14, 1, 7, 7, 是], [14, 2, 6, 6, 是], [14, 3, 5, 5, 是], [14, 4, 4, 4, 是], [14, 5, 3, 3, 是], [14, 6, 2, 2, 是], [14, 7, 1, 1, 是], [14, 8, 0, 0, 是], [14, 9, 21, 21, 是], [14, 10, 20, 20, 是], [14, 11, 19, 19, 是], [14, 12, 18, 18, 是], [14, 13, 17, 17, 是], [14, 14, 16, 16, 是], [14, 15, 15, 15, 是], [14, 16, 14, 14, 是], [14, 17, 13, 13, 是], [14, 18, 12, 12, 是], [14, 19, 11, 11, 是], [14, 20, 10, 10, 是], [14, 21, 9, 9, 是], [15, 0, 7, 7, 是], [15, 1, 6, 6, 是], [15, 2, 5, 5, 是], [15, 3, 4, 4, 是], [15, 4, 3, 3, 是], [15, 5, 2, 2, 是], [15, 6, 1, 1, 是], [15, 7, 0, 0, 是], [15, 8, 21, 21, 是], [15, 9, 20, 20, 是], [15, 10, 19, 19, 是], [15, 11, 18, 18, 是], [15, 12, 17, 17, 是], [15, 13, 16, 16, 是], [15, 14, 15, 15, 是], [15, 15, 14, 14, 是], [15, 16, 13, 13, 是], [15, 17, 12, 12, 是], [15, 18, 11, 11, 是], [15, 19, 10, 10, 是], [15, 20, 9, 9, 是], [15, 21, 8, 8, 是], [16, 0, 6, 6, 是], [16, 1, 5, 5, 是], [16, 2, 4, 4, 是], [16, 3, 3, 3, 是], [16, 4, 2, 2, 是], [16, 5, 1, 1, 是], [16, 6, 0, 0, 是], [16, 7, 21, 21, 是], [16, 8, 20, 20, 是], [16, 9, 19, 19, 是], [16, 10, 18, 18, 是], [16, 11, 17, 17, 是], [16, 12, 16, 16, 是], [16, 13, 15, 15, 是], [16, 14, 14, 14, 是], [16, 15, 13, 13, 是], [16, 16, 12, 12, 是], [16, 17, 11, 11, 是], [16, 18, 10, 10, 是], [16, 19, 9, 9, 是], [16, 20, 8, 8, 是], [16, 21, 7, 7, 是], [17, 0, 5, 5, 是], [17, 1, 4, 4, 是], [17, 2, 3, 3, 是], [17, 3, 2, 2, 是], [17, 4, 1, 1, 是], [17, 5, 0, 0, 是], [17, 6, 21, 21, 是], [17, 7, 20, 20, 是], [17, 8, 19, 19, 是], [17, 9, 18, 18, 是], [17, 10, 17, 17, 是], [17, 11, 16, 16, 是], [17, 12, 15, 15, 是], [17, 13, 14, 14, 是], [17, 14, 13, 13, 是], [17, 15, 12, 12, 是], [17, 16, 11, 11, 是], [17, 17, 10, 10, 是], [17, 18, 9, 9, 是], [17, 19, 8, 8, 是], [17, 20, 7, 7, 是], [17, 21, 6, 6, 是], [18, 0, 4, 4, 是], [18, 1, 3, 3, 是], [18, 2, 2, 2, 是], [18, 3, 1, 1, 是], [18, 4, 0, 0, 是], [18, 5, 21, 21, 是], [18, 6, 20, 20, 是], [18, 7, 19, 19, 是], [18, 8, 18, 18, 是], [18, 9, 17, 17, 是], [18, 10, 16, 16, 是], [18, 11, 15, 15, 是], [18, 12, 14, 14, 是], [18, 13, 13, 13, 是], [18, 14, 12, 12, 是], [18, 15, 11, 11, 是], [18, 16, 10, 10, 是], [18, 17, 9, 9, 是], [18, 18, 8, 8, 是], [18, 19, 7, 7, 是], [18, 20, 6, 6, 是], [18, 21, 5, 5, 是], [19, 0, 3, 3, 是], [19, 1, 2, 2, 是], [19, 2, 1, 1, 是], [19, 3, 0, 0, 是], [19, 4, 21, 21, 是], [19, 5, 20, 20, 是], [19, 6, 19, 19, 是], [19, 7, 18, 18, 是], [19, 8, 17, 17, 是], [19, 9, 16, 16, 是], [19, 10, 15, 15, 是], [19, 11, 14, 14, 是], [19, 12, 13, 13, 是], [19, 13, 12, 12, 是], [19, 14, 11, 11, 是], [19, 15, 10, 10, 是], [19, 16, 9, 9, 是], [19, 17, 8, 8, 是], [19, 18, 7, 7, 是], [19, 19, 6, 6, 是], [19, 20, 5, 5, 是], [19, 21, 4, 4, 是], [20, 0, 2, 2, 是], [20, 1, 1, 1, 是], [20, 2, 0, 0, 是], [20, 3, 21, 21, 是], [20, 4, 20, 20, 是], [20, 5, 19, 19, 是], [20, 6, 18, 18, 是], [20, 7, 17, 17, 是], [20, 8, 16, 16, 是], [20, 9, 15, 15, 是], [20, 10, 14, 14, 是], [20, 11, 13, 13, 是], [20, 12, 12, 12, 是], [20, 13, 11, 11, 是], [20, 14, 10, 10, 是], [20, 15, 9, 9, 是], [20, 16, 8, 8, 是], [20, 17, 7, 7, 是], [20, 18, 6, 6, 是], [20, 19, 5, 5, 是], [20, 20, 4, 4, 是], [20, 21, 3, 3, 是], [21, 0, 1, 1, 是], [21, 1, 0, 0, 是], [21, 2, 21, 21, 是], [21, 3, 20, 20, 是], [21, 4, 19, 19, 是], [21, 5, 18, 18, 是], [21, 6, 17, 17, 是], [21, 7, 16, 16, 是], [21, 8, 15, 15, 是], [21, 9, 14, 14, 是], [21, 10, 13, 13, 是], [21, 11, 12, 12, 是], [21, 12, 11, 11, 是], [21, 13, 10, 10, 是], [21, 14, 9, 9, 是], [21, 15, 8, 8, 是], [21, 16, 7, 7, 是], [21, 17, 6, 6, 是], [21, 18, 5, 5, 是], [21, 19, 4, 4, 是], [21, 20, 3, 3, 是], [21, 21, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141020,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],"target_identity":0,"mapping":[0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0021","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],\"target_identity\":0,\"mapping\":[0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 23, [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 22, 22, 是], [0, 2, 21, 21, 是], [0, 3, 20, 20, 是], [0, 4, 19, 19, 是], [0, 5, 18, 18, 是], [0, 6, 17, 17, 是], [0, 7, 16, 16, 是], [0, 8, 15, 15, 是], [0, 9, 14, 14, 是], [0, 10, 13, 13, 是], [0, 11, 12, 12, 是], [0, 12, 11, 11, 是], [0, 13, 10, 10, 是], [0, 14, 9, 9, 是], [0, 15, 8, 8, 是], [0, 16, 7, 7, 是], [0, 17, 6, 6, 是], [0, 18, 5, 5, 是], [0, 19, 4, 4, 是], [0, 20, 3, 3, 是], [0, 21, 2, 2, 是], [0, 22, 1, 1, 是], [1, 0, 22, 22, 是], [1, 1, 21, 21, 是], [1, 2, 20, 20, 是], [1, 3, 19, 19, 是], [1, 4, 18, 18, 是], [1, 5, 17, 17, 是], [1, 6, 16, 16, 是], [1, 7, 15, 15, 是], [1, 8, 14, 14, 是], [1, 9, 13, 13, 是], [1, 10, 12, 12, 是], [1, 11, 11, 11, 是], [1, 12, 10, 10, 是], [1, 13, 9, 9, 是], [1, 14, 8, 8, 是], [1, 15, 7, 7, 是], [1, 16, 6, 6, 是], [1, 17, 5, 5, 是], [1, 18, 4, 4, 是], [1, 19, 3, 3, 是], [1, 20, 2, 2, 是], [1, 21, 1, 1, 是], [1, 22, 0, 0, 是], [2, 0, 21, 21, 是], [2, 1, 20, 20, 是], [2, 2, 19, 19, 是], [2, 3, 18, 18, 是], [2, 4, 17, 17, 是], [2, 5, 16, 16, 是], [2, 6, 15, 15, 是], [2, 7, 14, 14, 是], [2, 8, 13, 13, 是], [2, 9, 12, 12, 是], [2, 10, 11, 11, 是], [2, 11, 10, 10, 是], [2, 12, 9, 9, 是], [2, 13, 8, 8, 是], [2, 14, 7, 7, 是], [2, 15, 6, 6, 是], [2, 16, 5, 5, 是], [2, 17, 4, 4, 是], [2, 18, 3, 3, 是], [2, 19, 2, 2, 是], [2, 20, 1, 1, 是], [2, 21, 0, 0, 是], [2, 22, 22, 22, 是], [3, 0, 20, 20, 是], [3, 1, 19, 19, 是], [3, 2, 18, 18, 是], [3, 3, 17, 17, 是], [3, 4, 16, 16, 是], [3, 5, 15, 15, 是], [3, 6, 14, 14, 是], [3, 7, 13, 13, 是], [3, 8, 12, 12, 是], [3, 9, 11, 11, 是], [3, 10, 10, 10, 是], [3, 11, 9, 9, 是], [3, 12, 8, 8, 是], [3, 13, 7, 7, 是], [3, 14, 6, 6, 是], [3, 15, 5, 5, 是], [3, 16, 4, 4, 是], [3, 17, 3, 3, 是], [3, 18, 2, 2, 是], [3, 19, 1, 1, 是], [3, 20, 0, 0, 是], [3, 21, 22, 22, 是], [3, 22, 21, 21, 是], [4, 0, 19, 19, 是], [4, 1, 18, 18, 是], [4, 2, 17, 17, 是], [4, 3, 16, 16, 是], [4, 4, 15, 15, 是], [4, 5, 14, 14, 是], [4, 6, 13, 13, 是], [4, 7, 12, 12, 是], [4, 8, 11, 11, 是], [4, 9, 10, 10, 是], [4, 10, 9, 9, 是], [4, 11, 8, 8, 是], [4, 12, 7, 7, 是], [4, 13, 6, 6, 是], [4, 14, 5, 5, 是], [4, 15, 4, 4, 是], [4, 16, 3, 3, 是], [4, 17, 2, 2, 是], [4, 18, 1, 1, 是], [4, 19, 0, 0, 是], [4, 20, 22, 22, 是], [4, 21, 21, 21, 是], [4, 22, 20, 20, 是], [5, 0, 18, 18, 是], [5, 1, 17, 17, 是], [5, 2, 16, 16, 是], [5, 3, 15, 15, 是], [5, 4, 14, 14, 是], [5, 5, 13, 13, 是], [5, 6, 12, 12, 是], [5, 7, 11, 11, 是], [5, 8, 10, 10, 是], [5, 9, 9, 9, 是], [5, 10, 8, 8, 是], [5, 11, 7, 7, 是], [5, 12, 6, 6, 是], [5, 13, 5, 5, 是], [5, 14, 4, 4, 是], [5, 15, 3, 3, 是], [5, 16, 2, 2, 是], [5, 17, 1, 1, 是], [5, 18, 0, 0, 是], [5, 19, 22, 22, 是], [5, 20, 21, 21, 是], [5, 21, 20, 20, 是], [5, 22, 19, 19, 是], [6, 0, 17, 17, 是], [6, 1, 16, 16, 是], [6, 2, 15, 15, 是], [6, 3, 14, 14, 是], [6, 4, 13, 13, 是], [6, 5, 12, 12, 是], [6, 6, 11, 11, 是], [6, 7, 10, 10, 是], [6, 8, 9, 9, 是], [6, 9, 8, 8, 是], [6, 10, 7, 7, 是], [6, 11, 6, 6, 是], [6, 12, 5, 5, 是], [6, 13, 4, 4, 是], [6, 14, 3, 3, 是], [6, 15, 2, 2, 是], [6, 16, 1, 1, 是], [6, 17, 0, 0, 是], [6, 18, 22, 22, 是], [6, 19, 21, 21, 是], [6, 20, 20, 20, 是], [6, 21, 19, 19, 是], [6, 22, 18, 18, 是], [7, 0, 16, 16, 是], [7, 1, 15, 15, 是], [7, 2, 14, 14, 是], [7, 3, 13, 13, 是], [7, 4, 12, 12, 是], [7, 5, 11, 11, 是], [7, 6, 10, 10, 是], [7, 7, 9, 9, 是], [7, 8, 8, 8, 是], [7, 9, 7, 7, 是], [7, 10, 6, 6, 是], [7, 11, 5, 5, 是], [7, 12, 4, 4, 是], [7, 13, 3, 3, 是], [7, 14, 2, 2, 是], [7, 15, 1, 1, 是], [7, 16, 0, 0, 是], [7, 17, 22, 22, 是], [7, 18, 21, 21, 是], [7, 19, 20, 20, 是], [7, 20, 19, 19, 是], [7, 21, 18, 18, 是], [7, 22, 17, 17, 是], [8, 0, 15, 15, 是], [8, 1, 14, 14, 是], [8, 2, 13, 13, 是], [8, 3, 12, 12, 是], [8, 4, 11, 11, 是], [8, 5, 10, 10, 是], [8, 6, 9, 9, 是], [8, 7, 8, 8, 是], [8, 8, 7, 7, 是], [8, 9, 6, 6, 是], [8, 10, 5, 5, 是], [8, 11, 4, 4, 是], [8, 12, 3, 3, 是], [8, 13, 2, 2, 是], [8, 14, 1, 1, 是], [8, 15, 0, 0, 是], [8, 16, 22, 22, 是], [8, 17, 21, 21, 是], [8, 18, 20, 20, 是], [8, 19, 19, 19, 是], [8, 20, 18, 18, 是], [8, 21, 17, 17, 是], [8, 22, 16, 16, 是], [9, 0, 14, 14, 是], [9, 1, 13, 13, 是], [9, 2, 12, 12, 是], [9, 3, 11, 11, 是], [9, 4, 10, 10, 是], [9, 5, 9, 9, 是], [9, 6, 8, 8, 是], [9, 7, 7, 7, 是], [9, 8, 6, 6, 是], [9, 9, 5, 5, 是], [9, 10, 4, 4, 是], [9, 11, 3, 3, 是], [9, 12, 2, 2, 是], [9, 13, 1, 1, 是], [9, 14, 0, 0, 是], [9, 15, 22, 22, 是], [9, 16, 21, 21, 是], [9, 17, 20, 20, 是], [9, 18, 19, 19, 是], [9, 19, 18, 18, 是], [9, 20, 17, 17, 是], [9, 21, 16, 16, 是], [9, 22, 15, 15, 是], [10, 0, 13, 13, 是], [10, 1, 12, 12, 是], [10, 2, 11, 11, 是], [10, 3, 10, 10, 是], [10, 4, 9, 9, 是], [10, 5, 8, 8, 是], [10, 6, 7, 7, 是], [10, 7, 6, 6, 是], [10, 8, 5, 5, 是], [10, 9, 4, 4, 是], [10, 10, 3, 3, 是], [10, 11, 2, 2, 是], [10, 12, 1, 1, 是], [10, 13, 0, 0, 是], [10, 14, 22, 22, 是], [10, 15, 21, 21, 是], [10, 16, 20, 20, 是], [10, 17, 19, 19, 是], [10, 18, 18, 18, 是], [10, 19, 17, 17, 是], [10, 20, 16, 16, 是], [10, 21, 15, 15, 是], [10, 22, 14, 14, 是], [11, 0, 12, 12, 是], [11, 1, 11, 11, 是], [11, 2, 10, 10, 是], [11, 3, 9, 9, 是], [11, 4, 8, 8, 是], [11, 5, 7, 7, 是], [11, 6, 6, 6, 是], [11, 7, 5, 5, 是], [11, 8, 4, 4, 是], [11, 9, 3, 3, 是], [11, 10, 2, 2, 是], [11, 11, 1, 1, 是], [11, 12, 0, 0, 是], [11, 13, 22, 22, 是], [11, 14, 21, 21, 是], [11, 15, 20, 20, 是], [11, 16, 19, 19, 是], [11, 17, 18, 18, 是], [11, 18, 17, 17, 是], [11, 19, 16, 16, 是], [11, 20, 15, 15, 是], [11, 21, 14, 14, 是], [11, 22, 13, 13, 是], [12, 0, 11, 11, 是], [12, 1, 10, 10, 是], [12, 2, 9, 9, 是], [12, 3, 8, 8, 是], [12, 4, 7, 7, 是], [12, 5, 6, 6, 是], [12, 6, 5, 5, 是], [12, 7, 4, 4, 是], [12, 8, 3, 3, 是], [12, 9, 2, 2, 是], [12, 10, 1, 1, 是], [12, 11, 0, 0, 是], [12, 12, 22, 22, 是], [12, 13, 21, 21, 是], [12, 14, 20, 20, 是], [12, 15, 19, 19, 是], [12, 16, 18, 18, 是], [12, 17, 17, 17, 是], [12, 18, 16, 16, 是], [12, 19, 15, 15, 是], [12, 20, 14, 14, 是], [12, 21, 13, 13, 是], [12, 22, 12, 12, 是], [13, 0, 10, 10, 是], [13, 1, 9, 9, 是], [13, 2, 8, 8, 是], [13, 3, 7, 7, 是], [13, 4, 6, 6, 是], [13, 5, 5, 5, 是], [13, 6, 4, 4, 是], [13, 7, 3, 3, 是], [13, 8, 2, 2, 是], [13, 9, 1, 1, 是], [13, 10, 0, 0, 是], [13, 11, 22, 22, 是], [13, 12, 21, 21, 是], [13, 13, 20, 20, 是], [13, 14, 19, 19, 是], [13, 15, 18, 18, 是], [13, 16, 17, 17, 是], [13, 17, 16, 16, 是], [13, 18, 15, 15, 是], [13, 19, 14, 14, 是], [13, 20, 13, 13, 是], [13, 21, 12, 12, 是], [13, 22, 11, 11, 是], [14, 0, 9, 9, 是], [14, 1, 8, 8, 是], [14, 2, 7, 7, 是], [14, 3, 6, 6, 是], [14, 4, 5, 5, 是], [14, 5, 4, 4, 是], [14, 6, 3, 3, 是], [14, 7, 2, 2, 是], [14, 8, 1, 1, 是], [14, 9, 0, 0, 是], [14, 10, 22, 22, 是], [14, 11, 21, 21, 是], [14, 12, 20, 20, 是], [14, 13, 19, 19, 是], [14, 14, 18, 18, 是], [14, 15, 17, 17, 是], [14, 16, 16, 16, 是], [14, 17, 15, 15, 是], [14, 18, 14, 14, 是], [14, 19, 13, 13, 是], [14, 20, 12, 12, 是], [14, 21, 11, 11, 是], [14, 22, 10, 10, 是], [15, 0, 8, 8, 是], [15, 1, 7, 7, 是], [15, 2, 6, 6, 是], [15, 3, 5, 5, 是], [15, 4, 4, 4, 是], [15, 5, 3, 3, 是], [15, 6, 2, 2, 是], [15, 7, 1, 1, 是], [15, 8, 0, 0, 是], [15, 9, 22, 22, 是], [15, 10, 21, 21, 是], [15, 11, 20, 20, 是], [15, 12, 19, 19, 是], [15, 13, 18, 18, 是], [15, 14, 17, 17, 是], [15, 15, 16, 16, 是], [15, 16, 15, 15, 是], [15, 17, 14, 14, 是], [15, 18, 13, 13, 是], [15, 19, 12, 12, 是], [15, 20, 11, 11, 是], [15, 21, 10, 10, 是], [15, 22, 9, 9, 是], [16, 0, 7, 7, 是], [16, 1, 6, 6, 是], [16, 2, 5, 5, 是], [16, 3, 4, 4, 是], [16, 4, 3, 3, 是], [16, 5, 2, 2, 是], [16, 6, 1, 1, 是], [16, 7, 0, 0, 是], [16, 8, 22, 22, 是], [16, 9, 21, 21, 是], [16, 10, 20, 20, 是], [16, 11, 19, 19, 是], [16, 12, 18, 18, 是], [16, 13, 17, 17, 是], [16, 14, 16, 16, 是], [16, 15, 15, 15, 是], [16, 16, 14, 14, 是], [16, 17, 13, 13, 是], [16, 18, 12, 12, 是], [16, 19, 11, 11, 是], [16, 20, 10, 10, 是], [16, 21, 9, 9, 是], [16, 22, 8, 8, 是], [17, 0, 6, 6, 是], [17, 1, 5, 5, 是], [17, 2, 4, 4, 是], [17, 3, 3, 3, 是], [17, 4, 2, 2, 是], [17, 5, 1, 1, 是], [17, 6, 0, 0, 是], [17, 7, 22, 22, 是], [17, 8, 21, 21, 是], [17, 9, 20, 20, 是], [17, 10, 19, 19, 是], [17, 11, 18, 18, 是], [17, 12, 17, 17, 是], [17, 13, 16, 16, 是], [17, 14, 15, 15, 是], [17, 15, 14, 14, 是], [17, 16, 13, 13, 是], [17, 17, 12, 12, 是], [17, 18, 11, 11, 是], [17, 19, 10, 10, 是], [17, 20, 9, 9, 是], [17, 21, 8, 8, 是], [17, 22, 7, 7, 是], [18, 0, 5, 5, 是], [18, 1, 4, 4, 是], [18, 2, 3, 3, 是], [18, 3, 2, 2, 是], [18, 4, 1, 1, 是], [18, 5, 0, 0, 是], [18, 6, 22, 22, 是], [18, 7, 21, 21, 是], [18, 8, 20, 20, 是], [18, 9, 19, 19, 是], [18, 10, 18, 18, 是], [18, 11, 17, 17, 是], [18, 12, 16, 16, 是], [18, 13, 15, 15, 是], [18, 14, 14, 14, 是], [18, 15, 13, 13, 是], [18, 16, 12, 12, 是], [18, 17, 11, 11, 是], [18, 18, 10, 10, 是], [18, 19, 9, 9, 是], [18, 20, 8, 8, 是], [18, 21, 7, 7, 是], [18, 22, 6, 6, 是], [19, 0, 4, 4, 是], [19, 1, 3, 3, 是], [19, 2, 2, 2, 是], [19, 3, 1, 1, 是], [19, 4, 0, 0, 是], [19, 5, 22, 22, 是], [19, 6, 21, 21, 是], [19, 7, 20, 20, 是], [19, 8, 19, 19, 是], [19, 9, 18, 18, 是], [19, 10, 17, 17, 是], [19, 11, 16, 16, 是], [19, 12, 15, 15, 是], [19, 13, 14, 14, 是], [19, 14, 13, 13, 是], [19, 15, 12, 12, 是], [19, 16, 11, 11, 是], [19, 17, 10, 10, 是], [19, 18, 9, 9, 是], [19, 19, 8, 8, 是], [19, 20, 7, 7, 是], [19, 21, 6, 6, 是], [19, 22, 5, 5, 是], [20, 0, 3, 3, 是], [20, 1, 2, 2, 是], [20, 2, 1, 1, 是], [20, 3, 0, 0, 是], [20, 4, 22, 22, 是], [20, 5, 21, 21, 是], [20, 6, 20, 20, 是], [20, 7, 19, 19, 是], [20, 8, 18, 18, 是], [20, 9, 17, 17, 是], [20, 10, 16, 16, 是], [20, 11, 15, 15, 是], [20, 12, 14, 14, 是], [20, 13, 13, 13, 是], [20, 14, 12, 12, 是], [20, 15, 11, 11, 是], [20, 16, 10, 10, 是], [20, 17, 9, 9, 是], [20, 18, 8, 8, 是], [20, 19, 7, 7, 是], [20, 20, 6, 6, 是], [20, 21, 5, 5, 是], [20, 22, 4, 4, 是], [21, 0, 2, 2, 是], [21, 1, 1, 1, 是], [21, 2, 0, 0, 是], [21, 3, 22, 22, 是], [21, 4, 21, 21, 是], [21, 5, 20, 20, 是], [21, 6, 19, 19, 是], [21, 7, 18, 18, 是], [21, 8, 17, 17, 是], [21, 9, 16, 16, 是], [21, 10, 15, 15, 是], [21, 11, 14, 14, 是], [21, 12, 13, 13, 是], [21, 13, 12, 12, 是], [21, 14, 11, 11, 是], [21, 15, 10, 10, 是], [21, 16, 9, 9, 是], [21, 17, 8, 8, 是], [21, 18, 7, 7, 是], [21, 19, 6, 6, 是], [21, 20, 5, 5, 是], [21, 21, 4, 4, 是], [21, 22, 3, 3, 是], [22, 0, 1, 1, 是], [22, 1, 0, 0, 是], [22, 2, 22, 22, 是], [22, 3, 21, 21, 是], [22, 4, 20, 20, 是], [22, 5, 19, 19, 是], [22, 6, 18, 18, 是], [22, 7, 17, 17, 是], [22, 8, 16, 16, 是], [22, 9, 15, 15, 是], [22, 10, 14, 14, 是], [22, 11, 13, 13, 是], [22, 12, 12, 12, 是], [22, 13, 11, 11, 是], [22, 14, 10, 10, 是], [22, 15, 9, 9, 是], [22, 16, 8, 8, 是], [22, 17, 7, 7, 是], [22, 18, 6, 6, 是], [22, 19, 5, 5, 是], [22, 20, 4, 4, 是], [22, 21, 3, 3, 是], [22, 22, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(23, 23, [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 22, 22, 是], [0, 2, 21, 21, 是], [0, 3, 20, 20, 是], [0, 4, 19, 19, 是], [0, 5, 18, 18, 是], [0, 6, 17, 17, 是], [0, 7, 16, 16, 是], [0, 8, 15, 15, 是], [0, 9, 14, 14, 是], [0, 10, 13, 13, 是], [0, 11, 12, 12, 是], [0, 12, 11, 11, 是], [0, 13, 10, 10, 是], [0, 14, 9, 9, 是], [0, 15, 8, 8, 是], [0, 16, 7, 7, 是], [0, 17, 6, 6, 是], [0, 18, 5, 5, 是], [0, 19, 4, 4, 是], [0, 20, 3, 3, 是], [0, 21, 2, 2, 是], [0, 22, 1, 1, 是], [1, 0, 22, 22, 是], [1, 1, 21, 21, 是], [1, 2, 20, 20, 是], [1, 3, 19, 19, 是], [1, 4, 18, 18, 是], [1, 5, 17, 17, 是], [1, 6, 16, 16, 是], [1, 7, 15, 15, 是], [1, 8, 14, 14, 是], [1, 9, 13, 13, 是], [1, 10, 12, 12, 是], [1, 11, 11, 11, 是], [1, 12, 10, 10, 是], [1, 13, 9, 9, 是], [1, 14, 8, 8, 是], [1, 15, 7, 7, 是], [1, 16, 6, 6, 是], [1, 17, 5, 5, 是], [1, 18, 4, 4, 是], [1, 19, 3, 3, 是], [1, 20, 2, 2, 是], [1, 21, 1, 1, 是], [1, 22, 0, 0, 是], [2, 0, 21, 21, 是], [2, 1, 20, 20, 是], [2, 2, 19, 19, 是], [2, 3, 18, 18, 是], [2, 4, 17, 17, 是], [2, 5, 16, 16, 是], [2, 6, 15, 15, 是], [2, 7, 14, 14, 是], [2, 8, 13, 13, 是], [2, 9, 12, 12, 是], [2, 10, 11, 11, 是], [2, 11, 10, 10, 是], [2, 12, 9, 9, 是], [2, 13, 8, 8, 是], [2, 14, 7, 7, 是], [2, 15, 6, 6, 是], [2, 16, 5, 5, 是], [2, 17, 4, 4, 是], [2, 18, 3, 3, 是], [2, 19, 2, 2, 是], [2, 20, 1, 1, 是], [2, 21, 0, 0, 是], [2, 22, 22, 22, 是], [3, 0, 20, 20, 是], [3, 1, 19, 19, 是], [3, 2, 18, 18, 是], [3, 3, 17, 17, 是], [3, 4, 16, 16, 是], [3, 5, 15, 15, 是], [3, 6, 14, 14, 是], [3, 7, 13, 13, 是], [3, 8, 12, 12, 是], [3, 9, 11, 11, 是], [3, 10, 10, 10, 是], [3, 11, 9, 9, 是], [3, 12, 8, 8, 是], [3, 13, 7, 7, 是], [3, 14, 6, 6, 是], [3, 15, 5, 5, 是], [3, 16, 4, 4, 是], [3, 17, 3, 3, 是], [3, 18, 2, 2, 是], [3, 19, 1, 1, 是], [3, 20, 0, 0, 是], [3, 21, 22, 22, 是], [3, 22, 21, 21, 是], [4, 0, 19, 19, 是], [4, 1, 18, 18, 是], [4, 2, 17, 17, 是], [4, 3, 16, 16, 是], [4, 4, 15, 15, 是], [4, 5, 14, 14, 是], [4, 6, 13, 13, 是], [4, 7, 12, 12, 是], [4, 8, 11, 11, 是], [4, 9, 10, 10, 是], [4, 10, 9, 9, 是], [4, 11, 8, 8, 是], [4, 12, 7, 7, 是], [4, 13, 6, 6, 是], [4, 14, 5, 5, 是], [4, 15, 4, 4, 是], [4, 16, 3, 3, 是], [4, 17, 2, 2, 是], [4, 18, 1, 1, 是], [4, 19, 0, 0, 是], [4, 20, 22, 22, 是], [4, 21, 21, 21, 是], [4, 22, 20, 20, 是], [5, 0, 18, 18, 是], [5, 1, 17, 17, 是], [5, 2, 16, 16, 是], [5, 3, 15, 15, 是], [5, 4, 14, 14, 是], [5, 5, 13, 13, 是], [5, 6, 12, 12, 是], [5, 7, 11, 11, 是], [5, 8, 10, 10, 是], [5, 9, 9, 9, 是], [5, 10, 8, 8, 是], [5, 11, 7, 7, 是], [5, 12, 6, 6, 是], [5, 13, 5, 5, 是], [5, 14, 4, 4, 是], [5, 15, 3, 3, 是], [5, 16, 2, 2, 是], [5, 17, 1, 1, 是], [5, 18, 0, 0, 是], [5, 19, 22, 22, 是], [5, 20, 21, 21, 是], [5, 21, 20, 20, 是], [5, 22, 19, 19, 是], [6, 0, 17, 17, 是], [6, 1, 16, 16, 是], [6, 2, 15, 15, 是], [6, 3, 14, 14, 是], [6, 4, 13, 13, 是], [6, 5, 12, 12, 是], [6, 6, 11, 11, 是], [6, 7, 10, 10, 是], [6, 8, 9, 9, 是], [6, 9, 8, 8, 是], [6, 10, 7, 7, 是], [6, 11, 6, 6, 是], [6, 12, 5, 5, 是], [6, 13, 4, 4, 是], [6, 14, 3, 3, 是], [6, 15, 2, 2, 是], [6, 16, 1, 1, 是], [6, 17, 0, 0, 是], [6, 18, 22, 22, 是], [6, 19, 21, 21, 是], [6, 20, 20, 20, 是], [6, 21, 19, 19, 是], [6, 22, 18, 18, 是], [7, 0, 16, 16, 是], [7, 1, 15, 15, 是], [7, 2, 14, 14, 是], [7, 3, 13, 13, 是], [7, 4, 12, 12, 是], [7, 5, 11, 11, 是], [7, 6, 10, 10, 是], [7, 7, 9, 9, 是], [7, 8, 8, 8, 是], [7, 9, 7, 7, 是], [7, 10, 6, 6, 是], [7, 11, 5, 5, 是], [7, 12, 4, 4, 是], [7, 13, 3, 3, 是], [7, 14, 2, 2, 是], [7, 15, 1, 1, 是], [7, 16, 0, 0, 是], [7, 17, 22, 22, 是], [7, 18, 21, 21, 是], [7, 19, 20, 20, 是], [7, 20, 19, 19, 是], [7, 21, 18, 18, 是], [7, 22, 17, 17, 是], [8, 0, 15, 15, 是], [8, 1, 14, 14, 是], [8, 2, 13, 13, 是], [8, 3, 12, 12, 是], [8, 4, 11, 11, 是], [8, 5, 10, 10, 是], [8, 6, 9, 9, 是], [8, 7, 8, 8, 是], [8, 8, 7, 7, 是], [8, 9, 6, 6, 是], [8, 10, 5, 5, 是], [8, 11, 4, 4, 是], [8, 12, 3, 3, 是], [8, 13, 2, 2, 是], [8, 14, 1, 1, 是], [8, 15, 0, 0, 是], [8, 16, 22, 22, 是], [8, 17, 21, 21, 是], [8, 18, 20, 20, 是], [8, 19, 19, 19, 是], [8, 20, 18, 18, 是], [8, 21, 17, 17, 是], [8, 22, 16, 16, 是], [9, 0, 14, 14, 是], [9, 1, 13, 13, 是], [9, 2, 12, 12, 是], [9, 3, 11, 11, 是], [9, 4, 10, 10, 是], [9, 5, 9, 9, 是], [9, 6, 8, 8, 是], [9, 7, 7, 7, 是], [9, 8, 6, 6, 是], [9, 9, 5, 5, 是], [9, 10, 4, 4, 是], [9, 11, 3, 3, 是], [9, 12, 2, 2, 是], [9, 13, 1, 1, 是], [9, 14, 0, 0, 是], [9, 15, 22, 22, 是], [9, 16, 21, 21, 是], [9, 17, 20, 20, 是], [9, 18, 19, 19, 是], [9, 19, 18, 18, 是], [9, 20, 17, 17, 是], [9, 21, 16, 16, 是], [9, 22, 15, 15, 是], [10, 0, 13, 13, 是], [10, 1, 12, 12, 是], [10, 2, 11, 11, 是], [10, 3, 10, 10, 是], [10, 4, 9, 9, 是], [10, 5, 8, 8, 是], [10, 6, 7, 7, 是], [10, 7, 6, 6, 是], [10, 8, 5, 5, 是], [10, 9, 4, 4, 是], [10, 10, 3, 3, 是], [10, 11, 2, 2, 是], [10, 12, 1, 1, 是], [10, 13, 0, 0, 是], [10, 14, 22, 22, 是], [10, 15, 21, 21, 是], [10, 16, 20, 20, 是], [10, 17, 19, 19, 是], [10, 18, 18, 18, 是], [10, 19, 17, 17, 是], [10, 20, 16, 16, 是], [10, 21, 15, 15, 是], [10, 22, 14, 14, 是], [11, 0, 12, 12, 是], [11, 1, 11, 11, 是], [11, 2, 10, 10, 是], [11, 3, 9, 9, 是], [11, 4, 8, 8, 是], [11, 5, 7, 7, 是], [11, 6, 6, 6, 是], [11, 7, 5, 5, 是], [11, 8, 4, 4, 是], [11, 9, 3, 3, 是], [11, 10, 2, 2, 是], [11, 11, 1, 1, 是], [11, 12, 0, 0, 是], [11, 13, 22, 22, 是], [11, 14, 21, 21, 是], [11, 15, 20, 20, 是], [11, 16, 19, 19, 是], [11, 17, 18, 18, 是], [11, 18, 17, 17, 是], [11, 19, 16, 16, 是], [11, 20, 15, 15, 是], [11, 21, 14, 14, 是], [11, 22, 13, 13, 是], [12, 0, 11, 11, 是], [12, 1, 10, 10, 是], [12, 2, 9, 9, 是], [12, 3, 8, 8, 是], [12, 4, 7, 7, 是], [12, 5, 6, 6, 是], [12, 6, 5, 5, 是], [12, 7, 4, 4, 是], [12, 8, 3, 3, 是], [12, 9, 2, 2, 是], [12, 10, 1, 1, 是], [12, 11, 0, 0, 是], [12, 12, 22, 22, 是], [12, 13, 21, 21, 是], [12, 14, 20, 20, 是], [12, 15, 19, 19, 是], [12, 16, 18, 18, 是], [12, 17, 17, 17, 是], [12, 18, 16, 16, 是], [12, 19, 15, 15, 是], [12, 20, 14, 14, 是], [12, 21, 13, 13, 是], [12, 22, 12, 12, 是], [13, 0, 10, 10, 是], [13, 1, 9, 9, 是], [13, 2, 8, 8, 是], [13, 3, 7, 7, 是], [13, 4, 6, 6, 是], [13, 5, 5, 5, 是], [13, 6, 4, 4, 是], [13, 7, 3, 3, 是], [13, 8, 2, 2, 是], [13, 9, 1, 1, 是], [13, 10, 0, 0, 是], [13, 11, 22, 22, 是], [13, 12, 21, 21, 是], [13, 13, 20, 20, 是], [13, 14, 19, 19, 是], [13, 15, 18, 18, 是], [13, 16, 17, 17, 是], [13, 17, 16, 16, 是], [13, 18, 15, 15, 是], [13, 19, 14, 14, 是], [13, 20, 13, 13, 是], [13, 21, 12, 12, 是], [13, 22, 11, 11, 是], [14, 0, 9, 9, 是], [14, 1, 8, 8, 是], [14, 2, 7, 7, 是], [14, 3, 6, 6, 是], [14, 4, 5, 5, 是], [14, 5, 4, 4, 是], [14, 6, 3, 3, 是], [14, 7, 2, 2, 是], [14, 8, 1, 1, 是], [14, 9, 0, 0, 是], [14, 10, 22, 22, 是], [14, 11, 21, 21, 是], [14, 12, 20, 20, 是], [14, 13, 19, 19, 是], [14, 14, 18, 18, 是], [14, 15, 17, 17, 是], [14, 16, 16, 16, 是], [14, 17, 15, 15, 是], [14, 18, 14, 14, 是], [14, 19, 13, 13, 是], [14, 20, 12, 12, 是], [14, 21, 11, 11, 是], [14, 22, 10, 10, 是], [15, 0, 8, 8, 是], [15, 1, 7, 7, 是], [15, 2, 6, 6, 是], [15, 3, 5, 5, 是], [15, 4, 4, 4, 是], [15, 5, 3, 3, 是], [15, 6, 2, 2, 是], [15, 7, 1, 1, 是], [15, 8, 0, 0, 是], [15, 9, 22, 22, 是], [15, 10, 21, 21, 是], [15, 11, 20, 20, 是], [15, 12, 19, 19, 是], [15, 13, 18, 18, 是], [15, 14, 17, 17, 是], [15, 15, 16, 16, 是], [15, 16, 15, 15, 是], [15, 17, 14, 14, 是], [15, 18, 13, 13, 是], [15, 19, 12, 12, 是], [15, 20, 11, 11, 是], [15, 21, 10, 10, 是], [15, 22, 9, 9, 是], [16, 0, 7, 7, 是], [16, 1, 6, 6, 是], [16, 2, 5, 5, 是], [16, 3, 4, 4, 是], [16, 4, 3, 3, 是], [16, 5, 2, 2, 是], [16, 6, 1, 1, 是], [16, 7, 0, 0, 是], [16, 8, 22, 22, 是], [16, 9, 21, 21, 是], [16, 10, 20, 20, 是], [16, 11, 19, 19, 是], [16, 12, 18, 18, 是], [16, 13, 17, 17, 是], [16, 14, 16, 16, 是], [16, 15, 15, 15, 是], [16, 16, 14, 14, 是], [16, 17, 13, 13, 是], [16, 18, 12, 12, 是], [16, 19, 11, 11, 是], [16, 20, 10, 10, 是], [16, 21, 9, 9, 是], [16, 22, 8, 8, 是], [17, 0, 6, 6, 是], [17, 1, 5, 5, 是], [17, 2, 4, 4, 是], [17, 3, 3, 3, 是], [17, 4, 2, 2, 是], [17, 5, 1, 1, 是], [17, 6, 0, 0, 是], [17, 7, 22, 22, 是], [17, 8, 21, 21, 是], [17, 9, 20, 20, 是], [17, 10, 19, 19, 是], [17, 11, 18, 18, 是], [17, 12, 17, 17, 是], [17, 13, 16, 16, 是], [17, 14, 15, 15, 是], [17, 15, 14, 14, 是], [17, 16, 13, 13, 是], [17, 17, 12, 12, 是], [17, 18, 11, 11, 是], [17, 19, 10, 10, 是], [17, 20, 9, 9, 是], [17, 21, 8, 8, 是], [17, 22, 7, 7, 是], [18, 0, 5, 5, 是], [18, 1, 4, 4, 是], [18, 2, 3, 3, 是], [18, 3, 2, 2, 是], [18, 4, 1, 1, 是], [18, 5, 0, 0, 是], [18, 6, 22, 22, 是], [18, 7, 21, 21, 是], [18, 8, 20, 20, 是], [18, 9, 19, 19, 是], [18, 10, 18, 18, 是], [18, 11, 17, 17, 是], [18, 12, 16, 16, 是], [18, 13, 15, 15, 是], [18, 14, 14, 14, 是], [18, 15, 13, 13, 是], [18, 16, 12, 12, 是], [18, 17, 11, 11, 是], [18, 18, 10, 10, 是], [18, 19, 9, 9, 是], [18, 20, 8, 8, 是], [18, 21, 7, 7, 是], [18, 22, 6, 6, 是], [19, 0, 4, 4, 是], [19, 1, 3, 3, 是], [19, 2, 2, 2, 是], [19, 3, 1, 1, 是], [19, 4, 0, 0, 是], [19, 5, 22, 22, 是], [19, 6, 21, 21, 是], [19, 7, 20, 20, 是], [19, 8, 19, 19, 是], [19, 9, 18, 18, 是], [19, 10, 17, 17, 是], [19, 11, 16, 16, 是], [19, 12, 15, 15, 是], [19, 13, 14, 14, 是], [19, 14, 13, 13, 是], [19, 15, 12, 12, 是], [19, 16, 11, 11, 是], [19, 17, 10, 10, 是], [19, 18, 9, 9, 是], [19, 19, 8, 8, 是], [19, 20, 7, 7, 是], [19, 21, 6, 6, 是], [19, 22, 5, 5, 是], [20, 0, 3, 3, 是], [20, 1, 2, 2, 是], [20, 2, 1, 1, 是], [20, 3, 0, 0, 是], [20, 4, 22, 22, 是], [20, 5, 21, 21, 是], [20, 6, 20, 20, 是], [20, 7, 19, 19, 是], [20, 8, 18, 18, 是], [20, 9, 17, 17, 是], [20, 10, 16, 16, 是], [20, 11, 15, 15, 是], [20, 12, 14, 14, 是], [20, 13, 13, 13, 是], [20, 14, 12, 12, 是], [20, 15, 11, 11, 是], [20, 16, 10, 10, 是], [20, 17, 9, 9, 是], [20, 18, 8, 8, 是], [20, 19, 7, 7, 是], [20, 20, 6, 6, 是], [20, 21, 5, 5, 是], [20, 22, 4, 4, 是], [21, 0, 2, 2, 是], [21, 1, 1, 1, 是], [21, 2, 0, 0, 是], [21, 3, 22, 22, 是], [21, 4, 21, 21, 是], [21, 5, 20, 20, 是], [21, 6, 19, 19, 是], [21, 7, 18, 18, 是], [21, 8, 17, 17, 是], [21, 9, 16, 16, 是], [21, 10, 15, 15, 是], [21, 11, 14, 14, 是], [21, 12, 13, 13, 是], [21, 13, 12, 12, 是], [21, 14, 11, 11, 是], [21, 15, 10, 10, 是], [21, 16, 9, 9, 是], [21, 17, 8, 8, 是], [21, 18, 7, 7, 是], [21, 19, 6, 6, 是], [21, 20, 5, 5, 是], [21, 21, 4, 4, 是], [21, 22, 3, 3, 是], [22, 0, 1, 1, 是], [22, 1, 0, 0, 是], [22, 2, 22, 22, 是], [22, 3, 21, 21, 是], [22, 4, 20, 20, 是], [22, 5, 19, 19, 是], [22, 6, 18, 18, 是], [22, 7, 17, 17, 是], [22, 8, 16, 16, 是], [22, 9, 15, 15, 是], [22, 10, 14, 14, 是], [22, 11, 13, 13, 是], [22, 12, 12, 12, 是], [22, 13, 11, 11, 是], [22, 14, 10, 10, 是], [22, 15, 9, 9, 是], [22, 16, 8, 8, 是], [22, 17, 7, 7, 是], [22, 18, 6, 6, 是], [22, 19, 5, 5, 是], [22, 20, 4, 4, 是], [22, 21, 3, 3, 是], [22, 22, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141021,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],"target_identity":0,"mapping":[0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0022","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"mapping\":[0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 24, [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 23, 23, 是], [0, 2, 22, 22, 是], [0, 3, 21, 21, 是], [0, 4, 20, 20, 是], [0, 5, 19, 19, 是], [0, 6, 18, 18, 是], [0, 7, 17, 17, 是], [0, 8, 16, 16, 是], [0, 9, 15, 15, 是], [0, 10, 14, 14, 是], [0, 11, 13, 13, 是], [0, 12, 12, 12, 是], [0, 13, 11, 11, 是], [0, 14, 10, 10, 是], [0, 15, 9, 9, 是], [0, 16, 8, 8, 是], [0, 17, 7, 7, 是], [0, 18, 6, 6, 是], [0, 19, 5, 5, 是], [0, 20, 4, 4, 是], [0, 21, 3, 3, 是], [0, 22, 2, 2, 是], [0, 23, 1, 1, 是], [1, 0, 23, 23, 是], [1, 1, 22, 22, 是], [1, 2, 21, 21, 是], [1, 3, 20, 20, 是], [1, 4, 19, 19, 是], [1, 5, 18, 18, 是], [1, 6, 17, 17, 是], [1, 7, 16, 16, 是], [1, 8, 15, 15, 是], [1, 9, 14, 14, 是], [1, 10, 13, 13, 是], [1, 11, 12, 12, 是], [1, 12, 11, 11, 是], [1, 13, 10, 10, 是], [1, 14, 9, 9, 是], [1, 15, 8, 8, 是], [1, 16, 7, 7, 是], [1, 17, 6, 6, 是], [1, 18, 5, 5, 是], [1, 19, 4, 4, 是], [1, 20, 3, 3, 是], [1, 21, 2, 2, 是], [1, 22, 1, 1, 是], [1, 23, 0, 0, 是], [2, 0, 22, 22, 是], [2, 1, 21, 21, 是], [2, 2, 20, 20, 是], [2, 3, 19, 19, 是], [2, 4, 18, 18, 是], [2, 5, 17, 17, 是], [2, 6, 16, 16, 是], [2, 7, 15, 15, 是], [2, 8, 14, 14, 是], [2, 9, 13, 13, 是], [2, 10, 12, 12, 是], [2, 11, 11, 11, 是], [2, 12, 10, 10, 是], [2, 13, 9, 9, 是], [2, 14, 8, 8, 是], [2, 15, 7, 7, 是], [2, 16, 6, 6, 是], [2, 17, 5, 5, 是], [2, 18, 4, 4, 是], [2, 19, 3, 3, 是], [2, 20, 2, 2, 是], [2, 21, 1, 1, 是], [2, 22, 0, 0, 是], [2, 23, 23, 23, 是], [3, 0, 21, 21, 是], [3, 1, 20, 20, 是], [3, 2, 19, 19, 是], [3, 3, 18, 18, 是], [3, 4, 17, 17, 是], [3, 5, 16, 16, 是], [3, 6, 15, 15, 是], [3, 7, 14, 14, 是], [3, 8, 13, 13, 是], [3, 9, 12, 12, 是], [3, 10, 11, 11, 是], [3, 11, 10, 10, 是], [3, 12, 9, 9, 是], [3, 13, 8, 8, 是], [3, 14, 7, 7, 是], [3, 15, 6, 6, 是], [3, 16, 5, 5, 是], [3, 17, 4, 4, 是], [3, 18, 3, 3, 是], [3, 19, 2, 2, 是], [3, 20, 1, 1, 是], [3, 21, 0, 0, 是], [3, 22, 23, 23, 是], [3, 23, 22, 22, 是], [4, 0, 20, 20, 是], [4, 1, 19, 19, 是], [4, 2, 18, 18, 是], [4, 3, 17, 17, 是], [4, 4, 16, 16, 是], [4, 5, 15, 15, 是], [4, 6, 14, 14, 是], [4, 7, 13, 13, 是], [4, 8, 12, 12, 是], [4, 9, 11, 11, 是], [4, 10, 10, 10, 是], [4, 11, 9, 9, 是], [4, 12, 8, 8, 是], [4, 13, 7, 7, 是], [4, 14, 6, 6, 是], [4, 15, 5, 5, 是], [4, 16, 4, 4, 是], [4, 17, 3, 3, 是], [4, 18, 2, 2, 是], [4, 19, 1, 1, 是], [4, 20, 0, 0, 是], [4, 21, 23, 23, 是], [4, 22, 22, 22, 是], [4, 23, 21, 21, 是], [5, 0, 19, 19, 是], [5, 1, 18, 18, 是], [5, 2, 17, 17, 是], [5, 3, 16, 16, 是], [5, 4, 15, 15, 是], [5, 5, 14, 14, 是], [5, 6, 13, 13, 是], [5, 7, 12, 12, 是], [5, 8, 11, 11, 是], [5, 9, 10, 10, 是], [5, 10, 9, 9, 是], [5, 11, 8, 8, 是], [5, 12, 7, 7, 是], [5, 13, 6, 6, 是], [5, 14, 5, 5, 是], [5, 15, 4, 4, 是], [5, 16, 3, 3, 是], [5, 17, 2, 2, 是], [5, 18, 1, 1, 是], [5, 19, 0, 0, 是], [5, 20, 23, 23, 是], [5, 21, 22, 22, 是], [5, 22, 21, 21, 是], [5, 23, 20, 20, 是], [6, 0, 18, 18, 是], [6, 1, 17, 17, 是], [6, 2, 16, 16, 是], [6, 3, 15, 15, 是], [6, 4, 14, 14, 是], [6, 5, 13, 13, 是], [6, 6, 12, 12, 是], [6, 7, 11, 11, 是], [6, 8, 10, 10, 是], [6, 9, 9, 9, 是], [6, 10, 8, 8, 是], [6, 11, 7, 7, 是], [6, 12, 6, 6, 是], [6, 13, 5, 5, 是], [6, 14, 4, 4, 是], [6, 15, 3, 3, 是], [6, 16, 2, 2, 是], [6, 17, 1, 1, 是], [6, 18, 0, 0, 是], [6, 19, 23, 23, 是], [6, 20, 22, 22, 是], [6, 21, 21, 21, 是], [6, 22, 20, 20, 是], [6, 23, 19, 19, 是], [7, 0, 17, 17, 是], [7, 1, 16, 16, 是], [7, 2, 15, 15, 是], [7, 3, 14, 14, 是], [7, 4, 13, 13, 是], [7, 5, 12, 12, 是], [7, 6, 11, 11, 是], [7, 7, 10, 10, 是], [7, 8, 9, 9, 是], [7, 9, 8, 8, 是], [7, 10, 7, 7, 是], [7, 11, 6, 6, 是], [7, 12, 5, 5, 是], [7, 13, 4, 4, 是], [7, 14, 3, 3, 是], [7, 15, 2, 2, 是], [7, 16, 1, 1, 是], [7, 17, 0, 0, 是], [7, 18, 23, 23, 是], [7, 19, 22, 22, 是], [7, 20, 21, 21, 是], [7, 21, 20, 20, 是], [7, 22, 19, 19, 是], [7, 23, 18, 18, 是], [8, 0, 16, 16, 是], [8, 1, 15, 15, 是], [8, 2, 14, 14, 是], [8, 3, 13, 13, 是], [8, 4, 12, 12, 是], [8, 5, 11, 11, 是], [8, 6, 10, 10, 是], [8, 7, 9, 9, 是], [8, 8, 8, 8, 是], [8, 9, 7, 7, 是], [8, 10, 6, 6, 是], [8, 11, 5, 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16, 22, 22, 是], [10, 17, 21, 21, 是], [10, 18, 20, 20, 是], [10, 19, 19, 19, 是], [10, 20, 18, 18, 是], [10, 21, 17, 17, 是], [10, 22, 16, 16, 是], [10, 23, 15, 15, 是], [11, 0, 13, 13, 是], [11, 1, 12, 12, 是], [11, 2, 11, 11, 是], [11, 3, 10, 10, 是], [11, 4, 9, 9, 是], [11, 5, 8, 8, 是], [11, 6, 7, 7, 是], [11, 7, 6, 6, 是], [11, 8, 5, 5, 是], [11, 9, 4, 4, 是], [11, 10, 3, 3, 是], [11, 11, 2, 2, 是], [11, 12, 1, 1, 是], [11, 13, 0, 0, 是], [11, 14, 23, 23, 是], [11, 15, 22, 22, 是], [11, 16, 21, 21, 是], [11, 17, 20, 20, 是], [11, 18, 19, 19, 是], [11, 19, 18, 18, 是], [11, 20, 17, 17, 是], [11, 21, 16, 16, 是], [11, 22, 15, 15, 是], [11, 23, 14, 14, 是], [12, 0, 12, 12, 是], [12, 1, 11, 11, 是], [12, 2, 10, 10, 是], [12, 3, 9, 9, 是], [12, 4, 8, 8, 是], [12, 5, 7, 7, 是], [12, 6, 6, 6, 是], [12, 7, 5, 5, 是], [12, 8, 4, 4, 是], [12, 9, 3, 3, 是], [12, 10, 2, 2, 是], [12, 11, 1, 1, 是], [12, 12, 0, 0, 是], [12, 13, 23, 23, 是], [12, 14, 22, 22, 是], [12, 15, 21, 21, 是], [12, 16, 20, 20, 是], [12, 17, 19, 19, 是], [12, 18, 18, 18, 是], [12, 19, 17, 17, 是], [12, 20, 16, 16, 是], [12, 21, 15, 15, 是], [12, 22, 14, 14, 是], [12, 23, 13, 13, 是], [13, 0, 11, 11, 是], [13, 1, 10, 10, 是], [13, 2, 9, 9, 是], [13, 3, 8, 8, 是], [13, 4, 7, 7, 是], [13, 5, 6, 6, 是], [13, 6, 5, 5, 是], [13, 7, 4, 4, 是], [13, 8, 3, 3, 是], [13, 9, 2, 2, 是], [13, 10, 1, 1, 是], [13, 11, 0, 0, 是], [13, 12, 23, 23, 是], [13, 13, 22, 22, 是], [13, 14, 21, 21, 是], [13, 15, 20, 20, 是], [13, 16, 19, 19, 是], [13, 17, 18, 18, 是], [13, 18, 17, 17, 是], [13, 19, 16, 16, 是], [13, 20, 15, 15, 是], [13, 21, 14, 14, 是], [13, 22, 13, 13, 是], [13, 23, 12, 12, 是], [14, 0, 10, 10, 是], [14, 1, 9, 9, 是], [14, 2, 8, 8, 是], [14, 3, 7, 7, 是], [14, 4, 6, 6, 是], [14, 5, 5, 5, 是], [14, 6, 4, 4, 是], [14, 7, 3, 3, 是], [14, 8, 2, 2, 是], [14, 9, 1, 1, 是], [14, 10, 0, 0, 是], [14, 11, 23, 23, 是], [14, 12, 22, 22, 是], [14, 13, 21, 21, 是], [14, 14, 20, 20, 是], [14, 15, 19, 19, 是], [14, 16, 18, 18, 是], [14, 17, 17, 17, 是], [14, 18, 16, 16, 是], [14, 19, 15, 15, 是], [14, 20, 14, 14, 是], [14, 21, 13, 13, 是], [14, 22, 12, 12, 是], [14, 23, 11, 11, 是], [15, 0, 9, 9, 是], [15, 1, 8, 8, 是], [15, 2, 7, 7, 是], [15, 3, 6, 6, 是], [15, 4, 5, 5, 是], [15, 5, 4, 4, 是], [15, 6, 3, 3, 是], [15, 7, 2, 2, 是], [15, 8, 1, 1, 是], [15, 9, 0, 0, 是], [15, 10, 23, 23, 是], [15, 11, 22, 22, 是], [15, 12, 21, 21, 是], [15, 13, 20, 20, 是], [15, 14, 19, 19, 是], [15, 15, 18, 18, 是], [15, 16, 17, 17, 是], [15, 17, 16, 16, 是], [15, 18, 15, 15, 是], [15, 19, 14, 14, 是], [15, 20, 13, 13, 是], [15, 21, 12, 12, 是], [15, 22, 11, 11, 是], [15, 23, 10, 10, 是], [16, 0, 8, 8, 是], [16, 1, 7, 7, 是], [16, 2, 6, 6, 是], [16, 3, 5, 5, 是], [16, 4, 4, 4, 是], [16, 5, 3, 3, 是], [16, 6, 2, 2, 是], [16, 7, 1, 1, 是], [16, 8, 0, 0, 是], [16, 9, 23, 23, 是], [16, 10, 22, 22, 是], [16, 11, 21, 21, 是], [16, 12, 20, 20, 是], [16, 13, 19, 19, 是], [16, 14, 18, 18, 是], [16, 15, 17, 17, 是], [16, 16, 16, 16, 是], [16, 17, 15, 15, 是], [16, 18, 14, 14, 是], [16, 19, 13, 13, 是], [16, 20, 12, 12, 是], [16, 21, 11, 11, 是], [16, 22, 10, 10, 是], [16, 23, 9, 9, 是], [17, 0, 7, 7, 是], [17, 1, 6, 6, 是], [17, 2, 5, 5, 是], [17, 3, 4, 4, 是], [17, 4, 3, 3, 是], [17, 5, 2, 2, 是], [17, 6, 1, 1, 是], [17, 7, 0, 0, 是], [17, 8, 23, 23, 是], [17, 9, 22, 22, 是], [17, 10, 21, 21, 是], [17, 11, 20, 20, 是], [17, 12, 19, 19, 是], [17, 13, 18, 18, 是], [17, 14, 17, 17, 是], [17, 15, 16, 16, 是], [17, 16, 15, 15, 是], [17, 17, 14, 14, 是], [17, 18, 13, 13, 是], [17, 19, 12, 12, 是], [17, 20, 11, 11, 是], [17, 21, 10, 10, 是], [17, 22, 9, 9, 是], [17, 23, 8, 8, 是], [18, 0, 6, 6, 是], [18, 1, 5, 5, 是], [18, 2, 4, 4, 是], [18, 3, 3, 3, 是], [18, 4, 2, 2, 是], [18, 5, 1, 1, 是], [18, 6, 0, 0, 是], [18, 7, 23, 23, 是], [18, 8, 22, 22, 是], [18, 9, 21, 21, 是], [18, 10, 20, 20, 是], [18, 11, 19, 19, 是], [18, 12, 18, 18, 是], [18, 13, 17, 17, 是], [18, 14, 16, 16, 是], [18, 15, 15, 15, 是], [18, 16, 14, 14, 是], [18, 17, 13, 13, 是], [18, 18, 12, 12, 是], [18, 19, 11, 11, 是], [18, 20, 10, 10, 是], [18, 21, 9, 9, 是], [18, 22, 8, 8, 是], [18, 23, 7, 7, 是], [19, 0, 5, 5, 是], [19, 1, 4, 4, 是], [19, 2, 3, 3, 是], [19, 3, 2, 2, 是], [19, 4, 1, 1, 是], [19, 5, 0, 0, 是], [19, 6, 23, 23, 是], [19, 7, 22, 22, 是], [19, 8, 21, 21, 是], [19, 9, 20, 20, 是], [19, 10, 19, 19, 是], [19, 11, 18, 18, 是], [19, 12, 17, 17, 是], [19, 13, 16, 16, 是], [19, 14, 15, 15, 是], [19, 15, 14, 14, 是], [19, 16, 13, 13, 是], [19, 17, 12, 12, 是], [19, 18, 11, 11, 是], [19, 19, 10, 10, 是], [19, 20, 9, 9, 是], [19, 21, 8, 8, 是], [19, 22, 7, 7, 是], [19, 23, 6, 6, 是], [20, 0, 4, 4, 是], [20, 1, 3, 3, 是], [20, 2, 2, 2, 是], [20, 3, 1, 1, 是], [20, 4, 0, 0, 是], [20, 5, 23, 23, 是], [20, 6, 22, 22, 是], [20, 7, 21, 21, 是], [20, 8, 20, 20, 是], [20, 9, 19, 19, 是], [20, 10, 18, 18, 是], [20, 11, 17, 17, 是], [20, 12, 16, 16, 是], [20, 13, 15, 15, 是], [20, 14, 14, 14, 是], [20, 15, 13, 13, 是], [20, 16, 12, 12, 是], [20, 17, 11, 11, 是], [20, 18, 10, 10, 是], [20, 19, 9, 9, 是], [20, 20, 8, 8, 是], [20, 21, 7, 7, 是], [20, 22, 6, 6, 是], [20, 23, 5, 5, 是], [21, 0, 3, 3, 是], [21, 1, 2, 2, 是], [21, 2, 1, 1, 是], [21, 3, 0, 0, 是], [21, 4, 23, 23, 是], [21, 5, 22, 22, 是], [21, 6, 21, 21, 是], [21, 7, 20, 20, 是], [21, 8, 19, 19, 是], [21, 9, 18, 18, 是], [21, 10, 17, 17, 是], [21, 11, 16, 16, 是], [21, 12, 15, 15, 是], [21, 13, 14, 14, 是], [21, 14, 13, 13, 是], [21, 15, 12, 12, 是], [21, 16, 11, 11, 是], [21, 17, 10, 10, 是], [21, 18, 9, 9, 是], [21, 19, 8, 8, 是], [21, 20, 7, 7, 是], [21, 21, 6, 6, 是], [21, 22, 5, 5, 是], [21, 23, 4, 4, 是], [22, 0, 2, 2, 是], [22, 1, 1, 1, 是], [22, 2, 0, 0, 是], [22, 3, 23, 23, 是], [22, 4, 22, 22, 是], [22, 5, 21, 21, 是], [22, 6, 20, 20, 是], [22, 7, 19, 19, 是], [22, 8, 18, 18, 是], [22, 9, 17, 17, 是], [22, 10, 16, 16, 是], [22, 11, 15, 15, 是], [22, 12, 14, 14, 是], [22, 13, 13, 13, 是], [22, 14, 12, 12, 是], [22, 15, 11, 11, 是], [22, 16, 10, 10, 是], [22, 17, 9, 9, 是], [22, 18, 8, 8, 是], [22, 19, 7, 7, 是], [22, 20, 6, 6, 是], [22, 21, 5, 5, 是], [22, 22, 4, 4, 是], [22, 23, 3, 3, 是], [23, 0, 1, 1, 是], [23, 1, 0, 0, 是], [23, 2, 23, 23, 是], [23, 3, 22, 22, 是], [23, 4, 21, 21, 是], [23, 5, 20, 20, 是], [23, 6, 19, 19, 是], [23, 7, 18, 18, 是], [23, 8, 17, 17, 是], [23, 9, 16, 16, 是], [23, 10, 15, 15, 是], [23, 11, 14, 14, 是], [23, 12, 13, 13, 是], [23, 13, 12, 12, 是], [23, 14, 11, 11, 是], [23, 15, 10, 10, 是], [23, 16, 9, 9, 是], [23, 17, 8, 8, 是], [23, 18, 7, 7, 是], [23, 19, 6, 6, 是], [23, 20, 5, 5, 是], [23, 21, 4, 4, 是], [23, 22, 3, 3, 是], [23, 23, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(24, 24, [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 23, 23, 是], [0, 2, 22, 22, 是], [0, 3, 21, 21, 是], [0, 4, 20, 20, 是], [0, 5, 19, 19, 是], [0, 6, 18, 18, 是], [0, 7, 17, 17, 是], [0, 8, 16, 16, 是], [0, 9, 15, 15, 是], [0, 10, 14, 14, 是], [0, 11, 13, 13, 是], [0, 12, 12, 12, 是], [0, 13, 11, 11, 是], [0, 14, 10, 10, 是], [0, 15, 9, 9, 是], [0, 16, 8, 8, 是], [0, 17, 7, 7, 是], [0, 18, 6, 6, 是], [0, 19, 5, 5, 是], [0, 20, 4, 4, 是], [0, 21, 3, 3, 是], [0, 22, 2, 2, 是], [0, 23, 1, 1, 是], [1, 0, 23, 23, 是], [1, 1, 22, 22, 是], [1, 2, 21, 21, 是], [1, 3, 20, 20, 是], [1, 4, 19, 19, 是], [1, 5, 18, 18, 是], [1, 6, 17, 17, 是], [1, 7, 16, 16, 是], [1, 8, 15, 15, 是], [1, 9, 14, 14, 是], [1, 10, 13, 13, 是], [1, 11, 12, 12, 是], [1, 12, 11, 11, 是], [1, 13, 10, 10, 是], [1, 14, 9, 9, 是], [1, 15, 8, 8, 是], [1, 16, 7, 7, 是], [1, 17, 6, 6, 是], [1, 18, 5, 5, 是], [1, 19, 4, 4, 是], [1, 20, 3, 3, 是], [1, 21, 2, 2, 是], [1, 22, 1, 1, 是], [1, 23, 0, 0, 是], [2, 0, 22, 22, 是], [2, 1, 21, 21, 是], [2, 2, 20, 20, 是], [2, 3, 19, 19, 是], [2, 4, 18, 18, 是], [2, 5, 17, 17, 是], [2, 6, 16, 16, 是], [2, 7, 15, 15, 是], [2, 8, 14, 14, 是], [2, 9, 13, 13, 是], [2, 10, 12, 12, 是], [2, 11, 11, 11, 是], [2, 12, 10, 10, 是], [2, 13, 9, 9, 是], [2, 14, 8, 8, 是], [2, 15, 7, 7, 是], [2, 16, 6, 6, 是], [2, 17, 5, 5, 是], [2, 18, 4, 4, 是], [2, 19, 3, 3, 是], [2, 20, 2, 2, 是], [2, 21, 1, 1, 是], [2, 22, 0, 0, 是], [2, 23, 23, 23, 是], [3, 0, 21, 21, 是], [3, 1, 20, 20, 是], [3, 2, 19, 19, 是], [3, 3, 18, 18, 是], [3, 4, 17, 17, 是], [3, 5, 16, 16, 是], [3, 6, 15, 15, 是], [3, 7, 14, 14, 是], [3, 8, 13, 13, 是], [3, 9, 12, 12, 是], [3, 10, 11, 11, 是], [3, 11, 10, 10, 是], [3, 12, 9, 9, 是], [3, 13, 8, 8, 是], [3, 14, 7, 7, 是], [3, 15, 6, 6, 是], [3, 16, 5, 5, 是], [3, 17, 4, 4, 是], [3, 18, 3, 3, 是], [3, 19, 2, 2, 是], [3, 20, 1, 1, 是], [3, 21, 0, 0, 是], [3, 22, 23, 23, 是], [3, 23, 22, 22, 是], [4, 0, 20, 20, 是], [4, 1, 19, 19, 是], [4, 2, 18, 18, 是], [4, 3, 17, 17, 是], [4, 4, 16, 16, 是], [4, 5, 15, 15, 是], [4, 6, 14, 14, 是], [4, 7, 13, 13, 是], [4, 8, 12, 12, 是], [4, 9, 11, 11, 是], [4, 10, 10, 10, 是], [4, 11, 9, 9, 是], [4, 12, 8, 8, 是], [4, 13, 7, 7, 是], [4, 14, 6, 6, 是], [4, 15, 5, 5, 是], [4, 16, 4, 4, 是], [4, 17, 3, 3, 是], [4, 18, 2, 2, 是], [4, 19, 1, 1, 是], [4, 20, 0, 0, 是], [4, 21, 23, 23, 是], [4, 22, 22, 22, 是], [4, 23, 21, 21, 是], [5, 0, 19, 19, 是], [5, 1, 18, 18, 是], [5, 2, 17, 17, 是], [5, 3, 16, 16, 是], [5, 4, 15, 15, 是], [5, 5, 14, 14, 是], [5, 6, 13, 13, 是], [5, 7, 12, 12, 是], [5, 8, 11, 11, 是], [5, 9, 10, 10, 是], [5, 10, 9, 9, 是], [5, 11, 8, 8, 是], [5, 12, 7, 7, 是], [5, 13, 6, 6, 是], [5, 14, 5, 5, 是], [5, 15, 4, 4, 是], [5, 16, 3, 3, 是], [5, 17, 2, 2, 是], [5, 18, 1, 1, 是], [5, 19, 0, 0, 是], [5, 20, 23, 23, 是], [5, 21, 22, 22, 是], [5, 22, 21, 21, 是], [5, 23, 20, 20, 是], [6, 0, 18, 18, 是], [6, 1, 17, 17, 是], [6, 2, 16, 16, 是], [6, 3, 15, 15, 是], [6, 4, 14, 14, 是], [6, 5, 13, 13, 是], [6, 6, 12, 12, 是], [6, 7, 11, 11, 是], [6, 8, 10, 10, 是], [6, 9, 9, 9, 是], [6, 10, 8, 8, 是], [6, 11, 7, 7, 是], [6, 12, 6, 6, 是], [6, 13, 5, 5, 是], [6, 14, 4, 4, 是], [6, 15, 3, 3, 是], [6, 16, 2, 2, 是], [6, 17, 1, 1, 是], [6, 18, 0, 0, 是], [6, 19, 23, 23, 是], [6, 20, 22, 22, 是], [6, 21, 21, 21, 是], [6, 22, 20, 20, 是], [6, 23, 19, 19, 是], [7, 0, 17, 17, 是], [7, 1, 16, 16, 是], [7, 2, 15, 15, 是], [7, 3, 14, 14, 是], [7, 4, 13, 13, 是], [7, 5, 12, 12, 是], [7, 6, 11, 11, 是], [7, 7, 10, 10, 是], [7, 8, 9, 9, 是], [7, 9, 8, 8, 是], [7, 10, 7, 7, 是], [7, 11, 6, 6, 是], [7, 12, 5, 5, 是], [7, 13, 4, 4, 是], [7, 14, 3, 3, 是], [7, 15, 2, 2, 是], [7, 16, 1, 1, 是], [7, 17, 0, 0, 是], [7, 18, 23, 23, 是], [7, 19, 22, 22, 是], [7, 20, 21, 21, 是], [7, 21, 20, 20, 是], [7, 22, 19, 19, 是], [7, 23, 18, 18, 是], [8, 0, 16, 16, 是], [8, 1, 15, 15, 是], [8, 2, 14, 14, 是], [8, 3, 13, 13, 是], [8, 4, 12, 12, 是], [8, 5, 11, 11, 是], [8, 6, 10, 10, 是], [8, 7, 9, 9, 是], [8, 8, 8, 8, 是], [8, 9, 7, 7, 是], [8, 10, 6, 6, 是], [8, 11, 5, 5, 是], [8, 12, 4, 4, 是], [8, 13, 3, 3, 是], [8, 14, 2, 2, 是], [8, 15, 1, 1, 是], [8, 16, 0, 0, 是], [8, 17, 23, 23, 是], [8, 18, 22, 22, 是], [8, 19, 21, 21, 是], [8, 20, 20, 20, 是], [8, 21, 19, 19, 是], [8, 22, 18, 18, 是], [8, 23, 17, 17, 是], [9, 0, 15, 15, 是], [9, 1, 14, 14, 是], [9, 2, 13, 13, 是], [9, 3, 12, 12, 是], [9, 4, 11, 11, 是], [9, 5, 10, 10, 是], [9, 6, 9, 9, 是], [9, 7, 8, 8, 是], [9, 8, 7, 7, 是], [9, 9, 6, 6, 是], [9, 10, 5, 5, 是], [9, 11, 4, 4, 是], [9, 12, 3, 3, 是], [9, 13, 2, 2, 是], [9, 14, 1, 1, 是], [9, 15, 0, 0, 是], [9, 16, 23, 23, 是], [9, 17, 22, 22, 是], [9, 18, 21, 21, 是], [9, 19, 20, 20, 是], [9, 20, 19, 19, 是], [9, 21, 18, 18, 是], [9, 22, 17, 17, 是], [9, 23, 16, 16, 是], [10, 0, 14, 14, 是], [10, 1, 13, 13, 是], [10, 2, 12, 12, 是], [10, 3, 11, 11, 是], [10, 4, 10, 10, 是], [10, 5, 9, 9, 是], [10, 6, 8, 8, 是], [10, 7, 7, 7, 是], [10, 8, 6, 6, 是], [10, 9, 5, 5, 是], [10, 10, 4, 4, 是], [10, 11, 3, 3, 是], [10, 12, 2, 2, 是], [10, 13, 1, 1, 是], [10, 14, 0, 0, 是], [10, 15, 23, 23, 是], [10, 16, 22, 22, 是], [10, 17, 21, 21, 是], [10, 18, 20, 20, 是], [10, 19, 19, 19, 是], [10, 20, 18, 18, 是], [10, 21, 17, 17, 是], [10, 22, 16, 16, 是], [10, 23, 15, 15, 是], [11, 0, 13, 13, 是], [11, 1, 12, 12, 是], [11, 2, 11, 11, 是], [11, 3, 10, 10, 是], [11, 4, 9, 9, 是], [11, 5, 8, 8, 是], [11, 6, 7, 7, 是], [11, 7, 6, 6, 是], [11, 8, 5, 5, 是], [11, 9, 4, 4, 是], [11, 10, 3, 3, 是], [11, 11, 2, 2, 是], [11, 12, 1, 1, 是], [11, 13, 0, 0, 是], [11, 14, 23, 23, 是], [11, 15, 22, 22, 是], [11, 16, 21, 21, 是], [11, 17, 20, 20, 是], [11, 18, 19, 19, 是], [11, 19, 18, 18, 是], [11, 20, 17, 17, 是], [11, 21, 16, 16, 是], [11, 22, 15, 15, 是], [11, 23, 14, 14, 是], [12, 0, 12, 12, 是], [12, 1, 11, 11, 是], [12, 2, 10, 10, 是], [12, 3, 9, 9, 是], [12, 4, 8, 8, 是], [12, 5, 7, 7, 是], [12, 6, 6, 6, 是], [12, 7, 5, 5, 是], [12, 8, 4, 4, 是], [12, 9, 3, 3, 是], [12, 10, 2, 2, 是], [12, 11, 1, 1, 是], [12, 12, 0, 0, 是], [12, 13, 23, 23, 是], [12, 14, 22, 22, 是], [12, 15, 21, 21, 是], [12, 16, 20, 20, 是], [12, 17, 19, 19, 是], [12, 18, 18, 18, 是], [12, 19, 17, 17, 是], [12, 20, 16, 16, 是], [12, 21, 15, 15, 是], [12, 22, 14, 14, 是], [12, 23, 13, 13, 是], [13, 0, 11, 11, 是], [13, 1, 10, 10, 是], [13, 2, 9, 9, 是], [13, 3, 8, 8, 是], [13, 4, 7, 7, 是], [13, 5, 6, 6, 是], [13, 6, 5, 5, 是], [13, 7, 4, 4, 是], [13, 8, 3, 3, 是], [13, 9, 2, 2, 是], [13, 10, 1, 1, 是], [13, 11, 0, 0, 是], [13, 12, 23, 23, 是], [13, 13, 22, 22, 是], [13, 14, 21, 21, 是], [13, 15, 20, 20, 是], [13, 16, 19, 19, 是], [13, 17, 18, 18, 是], [13, 18, 17, 17, 是], [13, 19, 16, 16, 是], [13, 20, 15, 15, 是], [13, 21, 14, 14, 是], [13, 22, 13, 13, 是], [13, 23, 12, 12, 是], [14, 0, 10, 10, 是], [14, 1, 9, 9, 是], [14, 2, 8, 8, 是], [14, 3, 7, 7, 是], [14, 4, 6, 6, 是], [14, 5, 5, 5, 是], [14, 6, 4, 4, 是], [14, 7, 3, 3, 是], [14, 8, 2, 2, 是], [14, 9, 1, 1, 是], [14, 10, 0, 0, 是], [14, 11, 23, 23, 是], [14, 12, 22, 22, 是], [14, 13, 21, 21, 是], [14, 14, 20, 20, 是], [14, 15, 19, 19, 是], [14, 16, 18, 18, 是], [14, 17, 17, 17, 是], [14, 18, 16, 16, 是], [14, 19, 15, 15, 是], [14, 20, 14, 14, 是], [14, 21, 13, 13, 是], [14, 22, 12, 12, 是], [14, 23, 11, 11, 是], [15, 0, 9, 9, 是], [15, 1, 8, 8, 是], [15, 2, 7, 7, 是], [15, 3, 6, 6, 是], [15, 4, 5, 5, 是], [15, 5, 4, 4, 是], [15, 6, 3, 3, 是], [15, 7, 2, 2, 是], [15, 8, 1, 1, 是], [15, 9, 0, 0, 是], [15, 10, 23, 23, 是], [15, 11, 22, 22, 是], [15, 12, 21, 21, 是], [15, 13, 20, 20, 是], [15, 14, 19, 19, 是], [15, 15, 18, 18, 是], [15, 16, 17, 17, 是], [15, 17, 16, 16, 是], [15, 18, 15, 15, 是], [15, 19, 14, 14, 是], [15, 20, 13, 13, 是], [15, 21, 12, 12, 是], [15, 22, 11, 11, 是], [15, 23, 10, 10, 是], [16, 0, 8, 8, 是], [16, 1, 7, 7, 是], [16, 2, 6, 6, 是], [16, 3, 5, 5, 是], [16, 4, 4, 4, 是], [16, 5, 3, 3, 是], [16, 6, 2, 2, 是], [16, 7, 1, 1, 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10, 20, 20, 是], [18, 11, 19, 19, 是], [18, 12, 18, 18, 是], [18, 13, 17, 17, 是], [18, 14, 16, 16, 是], [18, 15, 15, 15, 是], [18, 16, 14, 14, 是], [18, 17, 13, 13, 是], [18, 18, 12, 12, 是], [18, 19, 11, 11, 是], [18, 20, 10, 10, 是], [18, 21, 9, 9, 是], [18, 22, 8, 8, 是], [18, 23, 7, 7, 是], [19, 0, 5, 5, 是], [19, 1, 4, 4, 是], [19, 2, 3, 3, 是], [19, 3, 2, 2, 是], [19, 4, 1, 1, 是], [19, 5, 0, 0, 是], [19, 6, 23, 23, 是], [19, 7, 22, 22, 是], [19, 8, 21, 21, 是], [19, 9, 20, 20, 是], [19, 10, 19, 19, 是], [19, 11, 18, 18, 是], [19, 12, 17, 17, 是], [19, 13, 16, 16, 是], [19, 14, 15, 15, 是], [19, 15, 14, 14, 是], [19, 16, 13, 13, 是], [19, 17, 12, 12, 是], [19, 18, 11, 11, 是], [19, 19, 10, 10, 是], [19, 20, 9, 9, 是], [19, 21, 8, 8, 是], [19, 22, 7, 7, 是], [19, 23, 6, 6, 是], [20, 0, 4, 4, 是], [20, 1, 3, 3, 是], [20, 2, 2, 2, 是], [20, 3, 1, 1, 是], [20, 4, 0, 0, 是], [20, 5, 23, 23, 是], [20, 6, 22, 22, 是], [20, 7, 21, 21, 是], [20, 8, 20, 20, 是], [20, 9, 19, 19, 是], [20, 10, 18, 18, 是], [20, 11, 17, 17, 是], [20, 12, 16, 16, 是], [20, 13, 15, 15, 是], [20, 14, 14, 14, 是], [20, 15, 13, 13, 是], [20, 16, 12, 12, 是], [20, 17, 11, 11, 是], [20, 18, 10, 10, 是], [20, 19, 9, 9, 是], [20, 20, 8, 8, 是], [20, 21, 7, 7, 是], [20, 22, 6, 6, 是], [20, 23, 5, 5, 是], [21, 0, 3, 3, 是], [21, 1, 2, 2, 是], [21, 2, 1, 1, 是], [21, 3, 0, 0, 是], [21, 4, 23, 23, 是], [21, 5, 22, 22, 是], [21, 6, 21, 21, 是], [21, 7, 20, 20, 是], [21, 8, 19, 19, 是], [21, 9, 18, 18, 是], [21, 10, 17, 17, 是], [21, 11, 16, 16, 是], [21, 12, 15, 15, 是], [21, 13, 14, 14, 是], [21, 14, 13, 13, 是], [21, 15, 12, 12, 是], [21, 16, 11, 11, 是], [21, 17, 10, 10, 是], [21, 18, 9, 9, 是], [21, 19, 8, 8, 是], [21, 20, 7, 7, 是], [21, 21, 6, 6, 是], [21, 22, 5, 5, 是], [21, 23, 4, 4, 是], [22, 0, 2, 2, 是], [22, 1, 1, 1, 是], [22, 2, 0, 0, 是], [22, 3, 23, 23, 是], [22, 4, 22, 22, 是], [22, 5, 21, 21, 是], [22, 6, 20, 20, 是], [22, 7, 19, 19, 是], [22, 8, 18, 18, 是], [22, 9, 17, 17, 是], [22, 10, 16, 16, 是], [22, 11, 15, 15, 是], [22, 12, 14, 14, 是], [22, 13, 13, 13, 是], [22, 14, 12, 12, 是], [22, 15, 11, 11, 是], [22, 16, 10, 10, 是], [22, 17, 9, 9, 是], [22, 18, 8, 8, 是], [22, 19, 7, 7, 是], [22, 20, 6, 6, 是], [22, 21, 5, 5, 是], [22, 22, 4, 4, 是], [22, 23, 3, 3, 是], [23, 0, 1, 1, 是], [23, 1, 0, 0, 是], [23, 2, 23, 23, 是], [23, 3, 22, 22, 是], [23, 4, 21, 21, 是], [23, 5, 20, 20, 是], [23, 6, 19, 19, 是], [23, 7, 18, 18, 是], [23, 8, 17, 17, 是], [23, 9, 16, 16, 是], [23, 10, 15, 15, 是], [23, 11, 14, 14, 是], [23, 12, 13, 13, 是], [23, 13, 12, 12, 是], [23, 14, 11, 11, 是], [23, 15, 10, 10, 是], [23, 16, 9, 9, 是], [23, 17, 8, 8, 是], [23, 18, 7, 7, 是], [23, 19, 6, 6, 是], [23, 20, 5, 5, 是], [23, 21, 4, 4, 是], [23, 22, 3, 3, 是], [23, 23, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141022,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],"target_identity":0,"mapping":[0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0023","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],\"target_identity\":0,\"mapping\":[0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 25, [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 24, 24, 是], [0, 2, 23, 23, 是], [0, 3, 22, 22, 是], [0, 4, 21, 21, 是], [0, 5, 20, 20, 是], [0, 6, 19, 19, 是], [0, 7, 18, 18, 是], [0, 8, 17, 17, 是], [0, 9, 16, 16, 是], [0, 10, 15, 15, 是], [0, 11, 14, 14, 是], [0, 12, 13, 13, 是], [0, 13, 12, 12, 是], [0, 14, 11, 11, 是], [0, 15, 10, 10, 是], [0, 16, 9, 9, 是], [0, 17, 8, 8, 是], [0, 18, 7, 7, 是], [0, 19, 6, 6, 是], [0, 20, 5, 5, 是], [0, 21, 4, 4, 是], [0, 22, 3, 3, 是], [0, 23, 2, 2, 是], [0, 24, 1, 1, 是], [1, 0, 24, 24, 是], [1, 1, 23, 23, 是], [1, 2, 22, 22, 是], [1, 3, 21, 21, 是], [1, 4, 20, 20, 是], [1, 5, 19, 19, 是], [1, 6, 18, 18, 是], [1, 7, 17, 17, 是], [1, 8, 16, 16, 是], [1, 9, 15, 15, 是], [1, 10, 14, 14, 是], [1, 11, 13, 13, 是], [1, 12, 12, 12, 是], [1, 13, 11, 11, 是], [1, 14, 10, 10, 是], [1, 15, 9, 9, 是], [1, 16, 8, 8, 是], [1, 17, 7, 7, 是], [1, 18, 6, 6, 是], [1, 19, 5, 5, 是], [1, 20, 4, 4, 是], [1, 21, 3, 3, 是], [1, 22, 2, 2, 是], [1, 23, 1, 1, 是], [1, 24, 0, 0, 是], [2, 0, 23, 23, 是], [2, 1, 22, 22, 是], [2, 2, 21, 21, 是], [2, 3, 20, 20, 是], [2, 4, 19, 19, 是], [2, 5, 18, 18, 是], [2, 6, 17, 17, 是], [2, 7, 16, 16, 是], [2, 8, 15, 15, 是], [2, 9, 14, 14, 是], [2, 10, 13, 13, 是], [2, 11, 12, 12, 是], [2, 12, 11, 11, 是], [2, 13, 10, 10, 是], [2, 14, 9, 9, 是], [2, 15, 8, 8, 是], [2, 16, 7, 7, 是], [2, 17, 6, 6, 是], [2, 18, 5, 5, 是], [2, 19, 4, 4, 是], [2, 20, 3, 3, 是], [2, 21, 2, 2, 是], [2, 22, 1, 1, 是], [2, 23, 0, 0, 是], [2, 24, 24, 24, 是], [3, 0, 22, 22, 是], [3, 1, 21, 21, 是], [3, 2, 20, 20, 是], [3, 3, 19, 19, 是], [3, 4, 18, 18, 是], [3, 5, 17, 17, 是], [3, 6, 16, 16, 是], [3, 7, 15, 15, 是], [3, 8, 14, 14, 是], [3, 9, 13, 13, 是], [3, 10, 12, 12, 是], [3, 11, 11, 11, 是], [3, 12, 10, 10, 是], [3, 13, 9, 9, 是], [3, 14, 8, 8, 是], [3, 15, 7, 7, 是], [3, 16, 6, 6, 是], [3, 17, 5, 5, 是], [3, 18, 4, 4, 是], [3, 19, 3, 3, 是], [3, 20, 2, 2, 是], [3, 21, 1, 1, 是], [3, 22, 0, 0, 是], [3, 23, 24, 24, 是], [3, 24, 23, 23, 是], [4, 0, 21, 21, 是], [4, 1, 20, 20, 是], [4, 2, 19, 19, 是], [4, 3, 18, 18, 是], [4, 4, 17, 17, 是], [4, 5, 16, 16, 是], [4, 6, 15, 15, 是], [4, 7, 14, 14, 是], [4, 8, 13, 13, 是], [4, 9, 12, 12, 是], [4, 10, 11, 11, 是], [4, 11, 10, 10, 是], [4, 12, 9, 9, 是], [4, 13, 8, 8, 是], [4, 14, 7, 7, 是], [4, 15, 6, 6, 是], [4, 16, 5, 5, 是], [4, 17, 4, 4, 是], [4, 18, 3, 3, 是], [4, 19, 2, 2, 是], [4, 20, 1, 1, 是], [4, 21, 0, 0, 是], [4, 22, 24, 24, 是], [4, 23, 23, 23, 是], [4, 24, 22, 22, 是], [5, 0, 20, 20, 是], [5, 1, 19, 19, 是], [5, 2, 18, 18, 是], [5, 3, 17, 17, 是], [5, 4, 16, 16, 是], [5, 5, 15, 15, 是], [5, 6, 14, 14, 是], [5, 7, 13, 13, 是], [5, 8, 12, 12, 是], [5, 9, 11, 11, 是], [5, 10, 10, 10, 是], [5, 11, 9, 9, 是], [5, 12, 8, 8, 是], [5, 13, 7, 7, 是], [5, 14, 6, 6, 是], [5, 15, 5, 5, 是], [5, 16, 4, 4, 是], [5, 17, 3, 3, 是], [5, 18, 2, 2, 是], [5, 19, 1, 1, 是], [5, 20, 0, 0, 是], [5, 21, 24, 24, 是], [5, 22, 23, 23, 是], [5, 23, 22, 22, 是], [5, 24, 21, 21, 是], [6, 0, 19, 19, 是], [6, 1, 18, 18, 是], [6, 2, 17, 17, 是], [6, 3, 16, 16, 是], [6, 4, 15, 15, 是], [6, 5, 14, 14, 是], [6, 6, 13, 13, 是], [6, 7, 12, 12, 是], [6, 8, 11, 11, 是], [6, 9, 10, 10, 是], [6, 10, 9, 9, 是], [6, 11, 8, 8, 是], [6, 12, 7, 7, 是], [6, 13, 6, 6, 是], [6, 14, 5, 5, 是], [6, 15, 4, 4, 是], [6, 16, 3, 3, 是], [6, 17, 2, 2, 是], [6, 18, 1, 1, 是], [6, 19, 0, 0, 是], [6, 20, 24, 24, 是], [6, 21, 23, 23, 是], [6, 22, 22, 22, 是], [6, 23, 21, 21, 是], [6, 24, 20, 20, 是], [7, 0, 18, 18, 是], [7, 1, 17, 17, 是], [7, 2, 16, 16, 是], [7, 3, 15, 15, 是], [7, 4, 14, 14, 是], [7, 5, 13, 13, 是], [7, 6, 12, 12, 是], [7, 7, 11, 11, 是], [7, 8, 10, 10, 是], [7, 9, 9, 9, 是], [7, 10, 8, 8, 是], [7, 11, 7, 7, 是], [7, 12, 6, 6, 是], [7, 13, 5, 5, 是], [7, 14, 4, 4, 是], [7, 15, 3, 3, 是], [7, 16, 2, 2, 是], [7, 17, 1, 1, 是], [7, 18, 0, 0, 是], [7, 19, 24, 24, 是], [7, 20, 23, 23, 是], [7, 21, 22, 22, 是], [7, 22, 21, 21, 是], [7, 23, 20, 20, 是], [7, 24, 19, 19, 是], [8, 0, 17, 17, 是], [8, 1, 16, 16, 是], [8, 2, 15, 15, 是], [8, 3, 14, 14, 是], [8, 4, 13, 13, 是], [8, 5, 12, 12, 是], [8, 6, 11, 11, 是], [8, 7, 10, 10, 是], [8, 8, 9, 9, 是], [8, 9, 8, 8, 是], [8, 10, 7, 7, 是], [8, 11, 6, 6, 是], [8, 12, 5, 5, 是], [8, 13, 4, 4, 是], [8, 14, 3, 3, 是], [8, 15, 2, 2, 是], [8, 16, 1, 1, 是], [8, 17, 0, 0, 是], [8, 18, 24, 24, 是], [8, 19, 23, 23, 是], [8, 20, 22, 22, 是], [8, 21, 21, 21, 是], [8, 22, 20, 20, 是], [8, 23, 19, 19, 是], [8, 24, 18, 18, 是], [9, 0, 16, 16, 是], [9, 1, 15, 15, 是], [9, 2, 14, 14, 是], [9, 3, 13, 13, 是], [9, 4, 12, 12, 是], [9, 5, 11, 11, 是], [9, 6, 10, 10, 是], [9, 7, 9, 9, 是], [9, 8, 8, 8, 是], [9, 9, 7, 7, 是], [9, 10, 6, 6, 是], [9, 11, 5, 5, 是], [9, 12, 4, 4, 是], [9, 13, 3, 3, 是], [9, 14, 2, 2, 是], [9, 15, 1, 1, 是], [9, 16, 0, 0, 是], [9, 17, 24, 24, 是], [9, 18, 23, 23, 是], [9, 19, 22, 22, 是], [9, 20, 21, 21, 是], [9, 21, 20, 20, 是], [9, 22, 19, 19, 是], [9, 23, 18, 18, 是], [9, 24, 17, 17, 是], [10, 0, 15, 15, 是], [10, 1, 14, 14, 是], [10, 2, 13, 13, 是], [10, 3, 12, 12, 是], [10, 4, 11, 11, 是], [10, 5, 10, 10, 是], [10, 6, 9, 9, 是], [10, 7, 8, 8, 是], [10, 8, 7, 7, 是], [10, 9, 6, 6, 是], [10, 10, 5, 5, 是], [10, 11, 4, 4, 是], [10, 12, 3, 3, 是], [10, 13, 2, 2, 是], [10, 14, 1, 1, 是], [10, 15, 0, 0, 是], [10, 16, 24, 24, 是], [10, 17, 23, 23, 是], [10, 18, 22, 22, 是], [10, 19, 21, 21, 是], [10, 20, 20, 20, 是], [10, 21, 19, 19, 是], [10, 22, 18, 18, 是], [10, 23, 17, 17, 是], [10, 24, 16, 16, 是], [11, 0, 14, 14, 是], [11, 1, 13, 13, 是], [11, 2, 12, 12, 是], [11, 3, 11, 11, 是], [11, 4, 10, 10, 是], [11, 5, 9, 9, 是], [11, 6, 8, 8, 是], [11, 7, 7, 7, 是], [11, 8, 6, 6, 是], [11, 9, 5, 5, 是], [11, 10, 4, 4, 是], [11, 11, 3, 3, 是], [11, 12, 2, 2, 是], [11, 13, 1, 1, 是], [11, 14, 0, 0, 是], [11, 15, 24, 24, 是], [11, 16, 23, 23, 是], [11, 17, 22, 22, 是], [11, 18, 21, 21, 是], [11, 19, 20, 20, 是], [11, 20, 19, 19, 是], [11, 21, 18, 18, 是], [11, 22, 17, 17, 是], [11, 23, 16, 16, 是], [11, 24, 15, 15, 是], [12, 0, 13, 13, 是], [12, 1, 12, 12, 是], [12, 2, 11, 11, 是], [12, 3, 10, 10, 是], [12, 4, 9, 9, 是], [12, 5, 8, 8, 是], [12, 6, 7, 7, 是], [12, 7, 6, 6, 是], [12, 8, 5, 5, 是], [12, 9, 4, 4, 是], [12, 10, 3, 3, 是], [12, 11, 2, 2, 是], [12, 12, 1, 1, 是], [12, 13, 0, 0, 是], [12, 14, 24, 24, 是], [12, 15, 23, 23, 是], [12, 16, 22, 22, 是], [12, 17, 21, 21, 是], [12, 18, 20, 20, 是], [12, 19, 19, 19, 是], [12, 20, 18, 18, 是], [12, 21, 17, 17, 是], [12, 22, 16, 16, 是], [12, 23, 15, 15, 是], [12, 24, 14, 14, 是], [13, 0, 12, 12, 是], [13, 1, 11, 11, 是], [13, 2, 10, 10, 是], [13, 3, 9, 9, 是], [13, 4, 8, 8, 是], [13, 5, 7, 7, 是], [13, 6, 6, 6, 是], [13, 7, 5, 5, 是], [13, 8, 4, 4, 是], [13, 9, 3, 3, 是], [13, 10, 2, 2, 是], [13, 11, 1, 1, 是], [13, 12, 0, 0, 是], [13, 13, 24, 24, 是], [13, 14, 23, 23, 是], [13, 15, 22, 22, 是], [13, 16, 21, 21, 是], [13, 17, 20, 20, 是], [13, 18, 19, 19, 是], [13, 19, 18, 18, 是], [13, 20, 17, 17, 是], [13, 21, 16, 16, 是], [13, 22, 15, 15, 是], [13, 23, 14, 14, 是], [13, 24, 13, 13, 是], [14, 0, 11, 11, 是], [14, 1, 10, 10, 是], [14, 2, 9, 9, 是], [14, 3, 8, 8, 是], [14, 4, 7, 7, 是], [14, 5, 6, 6, 是], [14, 6, 5, 5, 是], [14, 7, 4, 4, 是], [14, 8, 3, 3, 是], [14, 9, 2, 2, 是], [14, 10, 1, 1, 是], [14, 11, 0, 0, 是], [14, 12, 24, 24, 是], [14, 13, 23, 23, 是], [14, 14, 22, 22, 是], [14, 15, 21, 21, 是], [14, 16, 20, 20, 是], [14, 17, 19, 19, 是], [14, 18, 18, 18, 是], [14, 19, 17, 17, 是], [14, 20, 16, 16, 是], [14, 21, 15, 15, 是], [14, 22, 14, 14, 是], [14, 23, 13, 13, 是], [14, 24, 12, 12, 是], [15, 0, 10, 10, 是], [15, 1, 9, 9, 是], [15, 2, 8, 8, 是], [15, 3, 7, 7, 是], [15, 4, 6, 6, 是], [15, 5, 5, 5, 是], [15, 6, 4, 4, 是], [15, 7, 3, 3, 是], [15, 8, 2, 2, 是], [15, 9, 1, 1, 是], [15, 10, 0, 0, 是], [15, 11, 24, 24, 是], [15, 12, 23, 23, 是], [15, 13, 22, 22, 是], [15, 14, 21, 21, 是], [15, 15, 20, 20, 是], [15, 16, 19, 19, 是], [15, 17, 18, 18, 是], [15, 18, 17, 17, 是], [15, 19, 16, 16, 是], [15, 20, 15, 15, 是], [15, 21, 14, 14, 是], [15, 22, 13, 13, 是], [15, 23, 12, 12, 是], [15, 24, 11, 11, 是], [16, 0, 9, 9, 是], [16, 1, 8, 8, 是], [16, 2, 7, 7, 是], [16, 3, 6, 6, 是], [16, 4, 5, 5, 是], [16, 5, 4, 4, 是], [16, 6, 3, 3, 是], [16, 7, 2, 2, 是], [16, 8, 1, 1, 是], [16, 9, 0, 0, 是], [16, 10, 24, 24, 是], [16, 11, 23, 23, 是], [16, 12, 22, 22, 是], [16, 13, 21, 21, 是], [16, 14, 20, 20, 是], [16, 15, 19, 19, 是], [16, 16, 18, 18, 是], [16, 17, 17, 17, 是], [16, 18, 16, 16, 是], [16, 19, 15, 15, 是], [16, 20, 14, 14, 是], [16, 21, 13, 13, 是], [16, 22, 12, 12, 是], [16, 23, 11, 11, 是], [16, 24, 10, 10, 是], [17, 0, 8, 8, 是], [17, 1, 7, 7, 是], [17, 2, 6, 6, 是], [17, 3, 5, 5, 是], [17, 4, 4, 4, 是], [17, 5, 3, 3, 是], [17, 6, 2, 2, 是], [17, 7, 1, 1, 是], [17, 8, 0, 0, 是], [17, 9, 24, 24, 是], [17, 10, 23, 23, 是], [17, 11, 22, 22, 是], [17, 12, 21, 21, 是], [17, 13, 20, 20, 是], [17, 14, 19, 19, 是], [17, 15, 18, 18, 是], [17, 16, 17, 17, 是], [17, 17, 16, 16, 是], [17, 18, 15, 15, 是], [17, 19, 14, 14, 是], [17, 20, 13, 13, 是], [17, 21, 12, 12, 是], [17, 22, 11, 11, 是], [17, 23, 10, 10, 是], [17, 24, 9, 9, 是], [18, 0, 7, 7, 是], [18, 1, 6, 6, 是], [18, 2, 5, 5, 是], [18, 3, 4, 4, 是], [18, 4, 3, 3, 是], [18, 5, 2, 2, 是], [18, 6, 1, 1, 是], [18, 7, 0, 0, 是], [18, 8, 24, 24, 是], [18, 9, 23, 23, 是], [18, 10, 22, 22, 是], [18, 11, 21, 21, 是], [18, 12, 20, 20, 是], [18, 13, 19, 19, 是], [18, 14, 18, 18, 是], [18, 15, 17, 17, 是], [18, 16, 16, 16, 是], [18, 17, 15, 15, 是], [18, 18, 14, 14, 是], [18, 19, 13, 13, 是], [18, 20, 12, 12, 是], [18, 21, 11, 11, 是], [18, 22, 10, 10, 是], [18, 23, 9, 9, 是], [18, 24, 8, 8, 是], [19, 0, 6, 6, 是], [19, 1, 5, 5, 是], [19, 2, 4, 4, 是], [19, 3, 3, 3, 是], [19, 4, 2, 2, 是], [19, 5, 1, 1, 是], [19, 6, 0, 0, 是], [19, 7, 24, 24, 是], [19, 8, 23, 23, 是], [19, 9, 22, 22, 是], [19, 10, 21, 21, 是], [19, 11, 20, 20, 是], [19, 12, 19, 19, 是], [19, 13, 18, 18, 是], [19, 14, 17, 17, 是], [19, 15, 16, 16, 是], [19, 16, 15, 15, 是], [19, 17, 14, 14, 是], [19, 18, 13, 13, 是], [19, 19, 12, 12, 是], [19, 20, 11, 11, 是], [19, 21, 10, 10, 是], [19, 22, 9, 9, 是], [19, 23, 8, 8, 是], [19, 24, 7, 7, 是], [20, 0, 5, 5, 是], [20, 1, 4, 4, 是], [20, 2, 3, 3, 是], [20, 3, 2, 2, 是], [20, 4, 1, 1, 是], [20, 5, 0, 0, 是], [20, 6, 24, 24, 是], [20, 7, 23, 23, 是], [20, 8, 22, 22, 是], [20, 9, 21, 21, 是], [20, 10, 20, 20, 是], [20, 11, 19, 19, 是], [20, 12, 18, 18, 是], [20, 13, 17, 17, 是], [20, 14, 16, 16, 是], [20, 15, 15, 15, 是], [20, 16, 14, 14, 是], [20, 17, 13, 13, 是], [20, 18, 12, 12, 是], [20, 19, 11, 11, 是], [20, 20, 10, 10, 是], [20, 21, 9, 9, 是], [20, 22, 8, 8, 是], [20, 23, 7, 7, 是], [20, 24, 6, 6, 是], [21, 0, 4, 4, 是], [21, 1, 3, 3, 是], [21, 2, 2, 2, 是], [21, 3, 1, 1, 是], [21, 4, 0, 0, 是], [21, 5, 24, 24, 是], [21, 6, 23, 23, 是], [21, 7, 22, 22, 是], [21, 8, 21, 21, 是], [21, 9, 20, 20, 是], [21, 10, 19, 19, 是], [21, 11, 18, 18, 是], [21, 12, 17, 17, 是], [21, 13, 16, 16, 是], [21, 14, 15, 15, 是], [21, 15, 14, 14, 是], [21, 16, 13, 13, 是], [21, 17, 12, 12, 是], [21, 18, 11, 11, 是], [21, 19, 10, 10, 是], [21, 20, 9, 9, 是], [21, 21, 8, 8, 是], [21, 22, 7, 7, 是], [21, 23, 6, 6, 是], [21, 24, 5, 5, 是], [22, 0, 3, 3, 是], [22, 1, 2, 2, 是], [22, 2, 1, 1, 是], [22, 3, 0, 0, 是], [22, 4, 24, 24, 是], [22, 5, 23, 23, 是], [22, 6, 22, 22, 是], [22, 7, 21, 21, 是], [22, 8, 20, 20, 是], [22, 9, 19, 19, 是], [22, 10, 18, 18, 是], [22, 11, 17, 17, 是], [22, 12, 16, 16, 是], [22, 13, 15, 15, 是], [22, 14, 14, 14, 是], [22, 15, 13, 13, 是], [22, 16, 12, 12, 是], [22, 17, 11, 11, 是], [22, 18, 10, 10, 是], [22, 19, 9, 9, 是], [22, 20, 8, 8, 是], [22, 21, 7, 7, 是], [22, 22, 6, 6, 是], [22, 23, 5, 5, 是], [22, 24, 4, 4, 是], [23, 0, 2, 2, 是], [23, 1, 1, 1, 是], [23, 2, 0, 0, 是], [23, 3, 24, 24, 是], [23, 4, 23, 23, 是], [23, 5, 22, 22, 是], [23, 6, 21, 21, 是], [23, 7, 20, 20, 是], [23, 8, 19, 19, 是], [23, 9, 18, 18, 是], [23, 10, 17, 17, 是], [23, 11, 16, 16, 是], [23, 12, 15, 15, 是], [23, 13, 14, 14, 是], [23, 14, 13, 13, 是], [23, 15, 12, 12, 是], [23, 16, 11, 11, 是], [23, 17, 10, 10, 是], [23, 18, 9, 9, 是], [23, 19, 8, 8, 是], [23, 20, 7, 7, 是], [23, 21, 6, 6, 是], [23, 22, 5, 5, 是], [23, 23, 4, 4, 是], [23, 24, 3, 3, 是], [24, 0, 1, 1, 是], [24, 1, 0, 0, 是], [24, 2, 24, 24, 是], [24, 3, 23, 23, 是], [24, 4, 22, 22, 是], [24, 5, 21, 21, 是], [24, 6, 20, 20, 是], [24, 7, 19, 19, 是], [24, 8, 18, 18, 是], [24, 9, 17, 17, 是], [24, 10, 16, 16, 是], [24, 11, 15, 15, 是], [24, 12, 14, 14, 是], [24, 13, 13, 13, 是], [24, 14, 12, 12, 是], [24, 15, 11, 11, 是], [24, 16, 10, 10, 是], [24, 17, 9, 9, 是], [24, 18, 8, 8, 是], [24, 19, 7, 7, 是], [24, 20, 6, 6, 是], [24, 21, 5, 5, 是], [24, 22, 4, 4, 是], [24, 23, 3, 3, 是], [24, 24, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(25, 25, [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 24, 24, 是], [0, 2, 23, 23, 是], [0, 3, 22, 22, 是], [0, 4, 21, 21, 是], [0, 5, 20, 20, 是], [0, 6, 19, 19, 是], [0, 7, 18, 18, 是], [0, 8, 17, 17, 是], [0, 9, 16, 16, 是], [0, 10, 15, 15, 是], [0, 11, 14, 14, 是], [0, 12, 13, 13, 是], [0, 13, 12, 12, 是], [0, 14, 11, 11, 是], [0, 15, 10, 10, 是], [0, 16, 9, 9, 是], [0, 17, 8, 8, 是], [0, 18, 7, 7, 是], [0, 19, 6, 6, 是], [0, 20, 5, 5, 是], [0, 21, 4, 4, 是], [0, 22, 3, 3, 是], [0, 23, 2, 2, 是], [0, 24, 1, 1, 是], [1, 0, 24, 24, 是], [1, 1, 23, 23, 是], [1, 2, 22, 22, 是], [1, 3, 21, 21, 是], [1, 4, 20, 20, 是], [1, 5, 19, 19, 是], [1, 6, 18, 18, 是], [1, 7, 17, 17, 是], [1, 8, 16, 16, 是], [1, 9, 15, 15, 是], [1, 10, 14, 14, 是], [1, 11, 13, 13, 是], [1, 12, 12, 12, 是], [1, 13, 11, 11, 是], [1, 14, 10, 10, 是], [1, 15, 9, 9, 是], [1, 16, 8, 8, 是], [1, 17, 7, 7, 是], [1, 18, 6, 6, 是], [1, 19, 5, 5, 是], [1, 20, 4, 4, 是], [1, 21, 3, 3, 是], [1, 22, 2, 2, 是], [1, 23, 1, 1, 是], [1, 24, 0, 0, 是], [2, 0, 23, 23, 是], [2, 1, 22, 22, 是], [2, 2, 21, 21, 是], [2, 3, 20, 20, 是], [2, 4, 19, 19, 是], [2, 5, 18, 18, 是], [2, 6, 17, 17, 是], [2, 7, 16, 16, 是], [2, 8, 15, 15, 是], [2, 9, 14, 14, 是], [2, 10, 13, 13, 是], [2, 11, 12, 12, 是], [2, 12, 11, 11, 是], [2, 13, 10, 10, 是], [2, 14, 9, 9, 是], [2, 15, 8, 8, 是], [2, 16, 7, 7, 是], [2, 17, 6, 6, 是], [2, 18, 5, 5, 是], [2, 19, 4, 4, 是], [2, 20, 3, 3, 是], [2, 21, 2, 2, 是], [2, 22, 1, 1, 是], [2, 23, 0, 0, 是], [2, 24, 24, 24, 是], [3, 0, 22, 22, 是], [3, 1, 21, 21, 是], [3, 2, 20, 20, 是], [3, 3, 19, 19, 是], [3, 4, 18, 18, 是], [3, 5, 17, 17, 是], [3, 6, 16, 16, 是], [3, 7, 15, 15, 是], [3, 8, 14, 14, 是], [3, 9, 13, 13, 是], [3, 10, 12, 12, 是], [3, 11, 11, 11, 是], [3, 12, 10, 10, 是], [3, 13, 9, 9, 是], [3, 14, 8, 8, 是], [3, 15, 7, 7, 是], [3, 16, 6, 6, 是], [3, 17, 5, 5, 是], [3, 18, 4, 4, 是], [3, 19, 3, 3, 是], [3, 20, 2, 2, 是], [3, 21, 1, 1, 是], [3, 22, 0, 0, 是], [3, 23, 24, 24, 是], [3, 24, 23, 23, 是], [4, 0, 21, 21, 是], [4, 1, 20, 20, 是], [4, 2, 19, 19, 是], [4, 3, 18, 18, 是], [4, 4, 17, 17, 是], [4, 5, 16, 16, 是], [4, 6, 15, 15, 是], [4, 7, 14, 14, 是], [4, 8, 13, 13, 是], [4, 9, 12, 12, 是], [4, 10, 11, 11, 是], [4, 11, 10, 10, 是], [4, 12, 9, 9, 是], [4, 13, 8, 8, 是], [4, 14, 7, 7, 是], [4, 15, 6, 6, 是], [4, 16, 5, 5, 是], [4, 17, 4, 4, 是], [4, 18, 3, 3, 是], [4, 19, 2, 2, 是], [4, 20, 1, 1, 是], [4, 21, 0, 0, 是], [4, 22, 24, 24, 是], [4, 23, 23, 23, 是], [4, 24, 22, 22, 是], [5, 0, 20, 20, 是], [5, 1, 19, 19, 是], [5, 2, 18, 18, 是], [5, 3, 17, 17, 是], [5, 4, 16, 16, 是], [5, 5, 15, 15, 是], [5, 6, 14, 14, 是], [5, 7, 13, 13, 是], [5, 8, 12, 12, 是], [5, 9, 11, 11, 是], [5, 10, 10, 10, 是], [5, 11, 9, 9, 是], [5, 12, 8, 8, 是], [5, 13, 7, 7, 是], [5, 14, 6, 6, 是], [5, 15, 5, 5, 是], [5, 16, 4, 4, 是], [5, 17, 3, 3, 是], [5, 18, 2, 2, 是], [5, 19, 1, 1, 是], [5, 20, 0, 0, 是], [5, 21, 24, 24, 是], [5, 22, 23, 23, 是], [5, 23, 22, 22, 是], [5, 24, 21, 21, 是], [6, 0, 19, 19, 是], [6, 1, 18, 18, 是], [6, 2, 17, 17, 是], [6, 3, 16, 16, 是], [6, 4, 15, 15, 是], [6, 5, 14, 14, 是], [6, 6, 13, 13, 是], [6, 7, 12, 12, 是], [6, 8, 11, 11, 是], [6, 9, 10, 10, 是], [6, 10, 9, 9, 是], [6, 11, 8, 8, 是], [6, 12, 7, 7, 是], [6, 13, 6, 6, 是], [6, 14, 5, 5, 是], [6, 15, 4, 4, 是], [6, 16, 3, 3, 是], [6, 17, 2, 2, 是], [6, 18, 1, 1, 是], [6, 19, 0, 0, 是], [6, 20, 24, 24, 是], [6, 21, 23, 23, 是], [6, 22, 22, 22, 是], [6, 23, 21, 21, 是], [6, 24, 20, 20, 是], [7, 0, 18, 18, 是], [7, 1, 17, 17, 是], [7, 2, 16, 16, 是], [7, 3, 15, 15, 是], [7, 4, 14, 14, 是], [7, 5, 13, 13, 是], [7, 6, 12, 12, 是], [7, 7, 11, 11, 是], [7, 8, 10, 10, 是], [7, 9, 9, 9, 是], [7, 10, 8, 8, 是], [7, 11, 7, 7, 是], [7, 12, 6, 6, 是], [7, 13, 5, 5, 是], [7, 14, 4, 4, 是], [7, 15, 3, 3, 是], [7, 16, 2, 2, 是], [7, 17, 1, 1, 是], [7, 18, 0, 0, 是], [7, 19, 24, 24, 是], [7, 20, 23, 23, 是], [7, 21, 22, 22, 是], [7, 22, 21, 21, 是], [7, 23, 20, 20, 是], [7, 24, 19, 19, 是], [8, 0, 17, 17, 是], [8, 1, 16, 16, 是], [8, 2, 15, 15, 是], [8, 3, 14, 14, 是], [8, 4, 13, 13, 是], [8, 5, 12, 12, 是], [8, 6, 11, 11, 是], [8, 7, 10, 10, 是], [8, 8, 9, 9, 是], [8, 9, 8, 8, 是], [8, 10, 7, 7, 是], [8, 11, 6, 6, 是], [8, 12, 5, 5, 是], [8, 13, 4, 4, 是], [8, 14, 3, 3, 是], [8, 15, 2, 2, 是], [8, 16, 1, 1, 是], [8, 17, 0, 0, 是], [8, 18, 24, 24, 是], [8, 19, 23, 23, 是], [8, 20, 22, 22, 是], [8, 21, 21, 21, 是], [8, 22, 20, 20, 是], [8, 23, 19, 19, 是], [8, 24, 18, 18, 是], [9, 0, 16, 16, 是], [9, 1, 15, 15, 是], [9, 2, 14, 14, 是], [9, 3, 13, 13, 是], [9, 4, 12, 12, 是], [9, 5, 11, 11, 是], [9, 6, 10, 10, 是], [9, 7, 9, 9, 是], [9, 8, 8, 8, 是], [9, 9, 7, 7, 是], [9, 10, 6, 6, 是], [9, 11, 5, 5, 是], [9, 12, 4, 4, 是], [9, 13, 3, 3, 是], [9, 14, 2, 2, 是], [9, 15, 1, 1, 是], [9, 16, 0, 0, 是], [9, 17, 24, 24, 是], [9, 18, 23, 23, 是], [9, 19, 22, 22, 是], [9, 20, 21, 21, 是], [9, 21, 20, 20, 是], [9, 22, 19, 19, 是], [9, 23, 18, 18, 是], [9, 24, 17, 17, 是], [10, 0, 15, 15, 是], [10, 1, 14, 14, 是], [10, 2, 13, 13, 是], [10, 3, 12, 12, 是], [10, 4, 11, 11, 是], [10, 5, 10, 10, 是], [10, 6, 9, 9, 是], [10, 7, 8, 8, 是], [10, 8, 7, 7, 是], [10, 9, 6, 6, 是], [10, 10, 5, 5, 是], [10, 11, 4, 4, 是], [10, 12, 3, 3, 是], [10, 13, 2, 2, 是], [10, 14, 1, 1, 是], [10, 15, 0, 0, 是], [10, 16, 24, 24, 是], [10, 17, 23, 23, 是], [10, 18, 22, 22, 是], [10, 19, 21, 21, 是], [10, 20, 20, 20, 是], [10, 21, 19, 19, 是], [10, 22, 18, 18, 是], [10, 23, 17, 17, 是], [10, 24, 16, 16, 是], [11, 0, 14, 14, 是], [11, 1, 13, 13, 是], [11, 2, 12, 12, 是], [11, 3, 11, 11, 是], [11, 4, 10, 10, 是], [11, 5, 9, 9, 是], [11, 6, 8, 8, 是], [11, 7, 7, 7, 是], [11, 8, 6, 6, 是], [11, 9, 5, 5, 是], [11, 10, 4, 4, 是], [11, 11, 3, 3, 是], [11, 12, 2, 2, 是], [11, 13, 1, 1, 是], [11, 14, 0, 0, 是], [11, 15, 24, 24, 是], [11, 16, 23, 23, 是], [11, 17, 22, 22, 是], [11, 18, 21, 21, 是], [11, 19, 20, 20, 是], [11, 20, 19, 19, 是], [11, 21, 18, 18, 是], [11, 22, 17, 17, 是], [11, 23, 16, 16, 是], [11, 24, 15, 15, 是], [12, 0, 13, 13, 是], [12, 1, 12, 12, 是], [12, 2, 11, 11, 是], [12, 3, 10, 10, 是], [12, 4, 9, 9, 是], [12, 5, 8, 8, 是], [12, 6, 7, 7, 是], [12, 7, 6, 6, 是], [12, 8, 5, 5, 是], [12, 9, 4, 4, 是], [12, 10, 3, 3, 是], [12, 11, 2, 2, 是], [12, 12, 1, 1, 是], [12, 13, 0, 0, 是], [12, 14, 24, 24, 是], [12, 15, 23, 23, 是], [12, 16, 22, 22, 是], [12, 17, 21, 21, 是], [12, 18, 20, 20, 是], [12, 19, 19, 19, 是], [12, 20, 18, 18, 是], [12, 21, 17, 17, 是], [12, 22, 16, 16, 是], [12, 23, 15, 15, 是], [12, 24, 14, 14, 是], [13, 0, 12, 12, 是], [13, 1, 11, 11, 是], [13, 2, 10, 10, 是], [13, 3, 9, 9, 是], [13, 4, 8, 8, 是], [13, 5, 7, 7, 是], [13, 6, 6, 6, 是], [13, 7, 5, 5, 是], [13, 8, 4, 4, 是], [13, 9, 3, 3, 是], [13, 10, 2, 2, 是], [13, 11, 1, 1, 是], [13, 12, 0, 0, 是], [13, 13, 24, 24, 是], [13, 14, 23, 23, 是], [13, 15, 22, 22, 是], [13, 16, 21, 21, 是], [13, 17, 20, 20, 是], [13, 18, 19, 19, 是], [13, 19, 18, 18, 是], [13, 20, 17, 17, 是], [13, 21, 16, 16, 是], [13, 22, 15, 15, 是], [13, 23, 14, 14, 是], [13, 24, 13, 13, 是], [14, 0, 11, 11, 是], [14, 1, 10, 10, 是], [14, 2, 9, 9, 是], [14, 3, 8, 8, 是], [14, 4, 7, 7, 是], [14, 5, 6, 6, 是], [14, 6, 5, 5, 是], [14, 7, 4, 4, 是], [14, 8, 3, 3, 是], [14, 9, 2, 2, 是], [14, 10, 1, 1, 是], [14, 11, 0, 0, 是], [14, 12, 24, 24, 是], [14, 13, 23, 23, 是], [14, 14, 22, 22, 是], [14, 15, 21, 21, 是], [14, 16, 20, 20, 是], [14, 17, 19, 19, 是], [14, 18, 18, 18, 是], [14, 19, 17, 17, 是], [14, 20, 16, 16, 是], [14, 21, 15, 15, 是], [14, 22, 14, 14, 是], [14, 23, 13, 13, 是], [14, 24, 12, 12, 是], [15, 0, 10, 10, 是], [15, 1, 9, 9, 是], [15, 2, 8, 8, 是], [15, 3, 7, 7, 是], [15, 4, 6, 6, 是], [15, 5, 5, 5, 是], [15, 6, 4, 4, 是], [15, 7, 3, 3, 是], [15, 8, 2, 2, 是], [15, 9, 1, 1, 是], [15, 10, 0, 0, 是], [15, 11, 24, 24, 是], [15, 12, 23, 23, 是], [15, 13, 22, 22, 是], [15, 14, 21, 21, 是], [15, 15, 20, 20, 是], [15, 16, 19, 19, 是], [15, 17, 18, 18, 是], [15, 18, 17, 17, 是], [15, 19, 16, 16, 是], [15, 20, 15, 15, 是], [15, 21, 14, 14, 是], [15, 22, 13, 13, 是], [15, 23, 12, 12, 是], [15, 24, 11, 11, 是], [16, 0, 9, 9, 是], [16, 1, 8, 8, 是], [16, 2, 7, 7, 是], [16, 3, 6, 6, 是], [16, 4, 5, 5, 是], [16, 5, 4, 4, 是], [16, 6, 3, 3, 是], [16, 7, 2, 2, 是], [16, 8, 1, 1, 是], [16, 9, 0, 0, 是], [16, 10, 24, 24, 是], [16, 11, 23, 23, 是], [16, 12, 22, 22, 是], [16, 13, 21, 21, 是], [16, 14, 20, 20, 是], [16, 15, 19, 19, 是], [16, 16, 18, 18, 是], [16, 17, 17, 17, 是], [16, 18, 16, 16, 是], [16, 19, 15, 15, 是], [16, 20, 14, 14, 是], [16, 21, 13, 13, 是], [16, 22, 12, 12, 是], [16, 23, 11, 11, 是], [16, 24, 10, 10, 是], [17, 0, 8, 8, 是], [17, 1, 7, 7, 是], [17, 2, 6, 6, 是], [17, 3, 5, 5, 是], [17, 4, 4, 4, 是], [17, 5, 3, 3, 是], [17, 6, 2, 2, 是], [17, 7, 1, 1, 是], [17, 8, 0, 0, 是], [17, 9, 24, 24, 是], [17, 10, 23, 23, 是], [17, 11, 22, 22, 是], [17, 12, 21, 21, 是], [17, 13, 20, 20, 是], [17, 14, 19, 19, 是], [17, 15, 18, 18, 是], [17, 16, 17, 17, 是], [17, 17, 16, 16, 是], [17, 18, 15, 15, 是], [17, 19, 14, 14, 是], [17, 20, 13, 13, 是], [17, 21, 12, 12, 是], [17, 22, 11, 11, 是], [17, 23, 10, 10, 是], [17, 24, 9, 9, 是], [18, 0, 7, 7, 是], [18, 1, 6, 6, 是], [18, 2, 5, 5, 是], [18, 3, 4, 4, 是], [18, 4, 3, 3, 是], [18, 5, 2, 2, 是], [18, 6, 1, 1, 是], [18, 7, 0, 0, 是], [18, 8, 24, 24, 是], [18, 9, 23, 23, 是], [18, 10, 22, 22, 是], [18, 11, 21, 21, 是], [18, 12, 20, 20, 是], [18, 13, 19, 19, 是], [18, 14, 18, 18, 是], [18, 15, 17, 17, 是], [18, 16, 16, 16, 是], [18, 17, 15, 15, 是], [18, 18, 14, 14, 是], [18, 19, 13, 13, 是], [18, 20, 12, 12, 是], [18, 21, 11, 11, 是], [18, 22, 10, 10, 是], [18, 23, 9, 9, 是], [18, 24, 8, 8, 是], [19, 0, 6, 6, 是], [19, 1, 5, 5, 是], [19, 2, 4, 4, 是], [19, 3, 3, 3, 是], [19, 4, 2, 2, 是], [19, 5, 1, 1, 是], [19, 6, 0, 0, 是], [19, 7, 24, 24, 是], [19, 8, 23, 23, 是], [19, 9, 22, 22, 是], [19, 10, 21, 21, 是], [19, 11, 20, 20, 是], [19, 12, 19, 19, 是], [19, 13, 18, 18, 是], [19, 14, 17, 17, 是], [19, 15, 16, 16, 是], [19, 16, 15, 15, 是], [19, 17, 14, 14, 是], [19, 18, 13, 13, 是], [19, 19, 12, 12, 是], [19, 20, 11, 11, 是], [19, 21, 10, 10, 是], [19, 22, 9, 9, 是], [19, 23, 8, 8, 是], [19, 24, 7, 7, 是], [20, 0, 5, 5, 是], [20, 1, 4, 4, 是], [20, 2, 3, 3, 是], [20, 3, 2, 2, 是], [20, 4, 1, 1, 是], [20, 5, 0, 0, 是], [20, 6, 24, 24, 是], [20, 7, 23, 23, 是], [20, 8, 22, 22, 是], [20, 9, 21, 21, 是], [20, 10, 20, 20, 是], [20, 11, 19, 19, 是], [20, 12, 18, 18, 是], [20, 13, 17, 17, 是], [20, 14, 16, 16, 是], [20, 15, 15, 15, 是], [20, 16, 14, 14, 是], [20, 17, 13, 13, 是], [20, 18, 12, 12, 是], [20, 19, 11, 11, 是], [20, 20, 10, 10, 是], [20, 21, 9, 9, 是], [20, 22, 8, 8, 是], [20, 23, 7, 7, 是], [20, 24, 6, 6, 是], [21, 0, 4, 4, 是], [21, 1, 3, 3, 是], [21, 2, 2, 2, 是], [21, 3, 1, 1, 是], [21, 4, 0, 0, 是], [21, 5, 24, 24, 是], [21, 6, 23, 23, 是], [21, 7, 22, 22, 是], [21, 8, 21, 21, 是], [21, 9, 20, 20, 是], [21, 10, 19, 19, 是], [21, 11, 18, 18, 是], [21, 12, 17, 17, 是], [21, 13, 16, 16, 是], [21, 14, 15, 15, 是], [21, 15, 14, 14, 是], [21, 16, 13, 13, 是], [21, 17, 12, 12, 是], [21, 18, 11, 11, 是], [21, 19, 10, 10, 是], [21, 20, 9, 9, 是], [21, 21, 8, 8, 是], [21, 22, 7, 7, 是], [21, 23, 6, 6, 是], [21, 24, 5, 5, 是], [22, 0, 3, 3, 是], [22, 1, 2, 2, 是], [22, 2, 1, 1, 是], [22, 3, 0, 0, 是], [22, 4, 24, 24, 是], [22, 5, 23, 23, 是], [22, 6, 22, 22, 是], [22, 7, 21, 21, 是], [22, 8, 20, 20, 是], [22, 9, 19, 19, 是], [22, 10, 18, 18, 是], [22, 11, 17, 17, 是], [22, 12, 16, 16, 是], [22, 13, 15, 15, 是], [22, 14, 14, 14, 是], [22, 15, 13, 13, 是], [22, 16, 12, 12, 是], [22, 17, 11, 11, 是], [22, 18, 10, 10, 是], [22, 19, 9, 9, 是], [22, 20, 8, 8, 是], [22, 21, 7, 7, 是], [22, 22, 6, 6, 是], [22, 23, 5, 5, 是], [22, 24, 4, 4, 是], [23, 0, 2, 2, 是], [23, 1, 1, 1, 是], [23, 2, 0, 0, 是], [23, 3, 24, 24, 是], [23, 4, 23, 23, 是], [23, 5, 22, 22, 是], [23, 6, 21, 21, 是], [23, 7, 20, 20, 是], [23, 8, 19, 19, 是], [23, 9, 18, 18, 是], [23, 10, 17, 17, 是], [23, 11, 16, 16, 是], [23, 12, 15, 15, 是], [23, 13, 14, 14, 是], [23, 14, 13, 13, 是], [23, 15, 12, 12, 是], [23, 16, 11, 11, 是], [23, 17, 10, 10, 是], [23, 18, 9, 9, 是], [23, 19, 8, 8, 是], [23, 20, 7, 7, 是], [23, 21, 6, 6, 是], [23, 22, 5, 5, 是], [23, 23, 4, 4, 是], [23, 24, 3, 3, 是], [24, 0, 1, 1, 是], [24, 1, 0, 0, 是], [24, 2, 24, 24, 是], [24, 3, 23, 23, 是], [24, 4, 22, 22, 是], [24, 5, 21, 21, 是], [24, 6, 20, 20, 是], [24, 7, 19, 19, 是], [24, 8, 18, 18, 是], [24, 9, 17, 17, 是], [24, 10, 16, 16, 是], [24, 11, 15, 15, 是], [24, 12, 14, 14, 是], [24, 13, 13, 13, 是], [24, 14, 12, 12, 是], [24, 15, 11, 11, 是], [24, 16, 10, 10, 是], [24, 17, 9, 9, 是], [24, 18, 8, 8, 是], [24, 19, 7, 7, 是], [24, 20, 6, 6, 是], [24, 21, 5, 5, 是], [24, 22, 4, 4, 是], [24, 23, 3, 3, 是], [24, 24, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141023,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],"target_identity":0,"mapping":[0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0024","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"mapping\":[0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(26, 26, [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 25, 25, 是], [0, 2, 24, 24, 是], [0, 3, 23, 23, 是], [0, 4, 22, 22, 是], [0, 5, 21, 21, 是], [0, 6, 20, 20, 是], [0, 7, 19, 19, 是], [0, 8, 18, 18, 是], [0, 9, 17, 17, 是], [0, 10, 16, 16, 是], [0, 11, 15, 15, 是], [0, 12, 14, 14, 是], [0, 13, 13, 13, 是], [0, 14, 12, 12, 是], [0, 15, 11, 11, 是], [0, 16, 10, 10, 是], [0, 17, 9, 9, 是], [0, 18, 8, 8, 是], [0, 19, 7, 7, 是], [0, 20, 6, 6, 是], [0, 21, 5, 5, 是], [0, 22, 4, 4, 是], [0, 23, 3, 3, 是], [0, 24, 2, 2, 是], [0, 25, 1, 1, 是], [1, 0, 25, 25, 是], [1, 1, 24, 24, 是], [1, 2, 23, 23, 是], [1, 3, 22, 22, 是], [1, 4, 21, 21, 是], [1, 5, 20, 20, 是], [1, 6, 19, 19, 是], [1, 7, 18, 18, 是], [1, 8, 17, 17, 是], [1, 9, 16, 16, 是], [1, 10, 15, 15, 是], [1, 11, 14, 14, 是], [1, 12, 13, 13, 是], [1, 13, 12, 12, 是], [1, 14, 11, 11, 是], [1, 15, 10, 10, 是], [1, 16, 9, 9, 是], [1, 17, 8, 8, 是], [1, 18, 7, 7, 是], [1, 19, 6, 6, 是], [1, 20, 5, 5, 是], [1, 21, 4, 4, 是], [1, 22, 3, 3, 是], [1, 23, 2, 2, 是], [1, 24, 1, 1, 是], [1, 25, 0, 0, 是], [2, 0, 24, 24, 是], [2, 1, 23, 23, 是], [2, 2, 22, 22, 是], [2, 3, 21, 21, 是], [2, 4, 20, 20, 是], [2, 5, 19, 19, 是], [2, 6, 18, 18, 是], [2, 7, 17, 17, 是], [2, 8, 16, 16, 是], [2, 9, 15, 15, 是], [2, 10, 14, 14, 是], [2, 11, 13, 13, 是], [2, 12, 12, 12, 是], [2, 13, 11, 11, 是], [2, 14, 10, 10, 是], [2, 15, 9, 9, 是], [2, 16, 8, 8, 是], [2, 17, 7, 7, 是], [2, 18, 6, 6, 是], [2, 19, 5, 5, 是], [2, 20, 4, 4, 是], [2, 21, 3, 3, 是], [2, 22, 2, 2, 是], [2, 23, 1, 1, 是], [2, 24, 0, 0, 是], [2, 25, 25, 25, 是], [3, 0, 23, 23, 是], [3, 1, 22, 22, 是], [3, 2, 21, 21, 是], [3, 3, 20, 20, 是], [3, 4, 19, 19, 是], [3, 5, 18, 18, 是], [3, 6, 17, 17, 是], [3, 7, 16, 16, 是], [3, 8, 15, 15, 是], [3, 9, 14, 14, 是], [3, 10, 13, 13, 是], [3, 11, 12, 12, 是], [3, 12, 11, 11, 是], [3, 13, 10, 10, 是], [3, 14, 9, 9, 是], [3, 15, 8, 8, 是], [3, 16, 7, 7, 是], [3, 17, 6, 6, 是], [3, 18, 5, 5, 是], [3, 19, 4, 4, 是], [3, 20, 3, 3, 是], [3, 21, 2, 2, 是], [3, 22, 1, 1, 是], [3, 23, 0, 0, 是], [3, 24, 25, 25, 是], [3, 25, 24, 24, 是], [4, 0, 22, 22, 是], [4, 1, 21, 21, 是], [4, 2, 20, 20, 是], [4, 3, 19, 19, 是], [4, 4, 18, 18, 是], [4, 5, 17, 17, 是], [4, 6, 16, 16, 是], [4, 7, 15, 15, 是], [4, 8, 14, 14, 是], [4, 9, 13, 13, 是], [4, 10, 12, 12, 是], [4, 11, 11, 11, 是], [4, 12, 10, 10, 是], [4, 13, 9, 9, 是], [4, 14, 8, 8, 是], [4, 15, 7, 7, 是], [4, 16, 6, 6, 是], [4, 17, 5, 5, 是], [4, 18, 4, 4, 是], [4, 19, 3, 3, 是], [4, 20, 2, 2, 是], [4, 21, 1, 1, 是], [4, 22, 0, 0, 是], [4, 23, 25, 25, 是], [4, 24, 24, 24, 是], [4, 25, 23, 23, 是], [5, 0, 21, 21, 是], [5, 1, 20, 20, 是], [5, 2, 19, 19, 是], [5, 3, 18, 18, 是], [5, 4, 17, 17, 是], [5, 5, 16, 16, 是], [5, 6, 15, 15, 是], [5, 7, 14, 14, 是], [5, 8, 13, 13, 是], [5, 9, 12, 12, 是], [5, 10, 11, 11, 是], [5, 11, 10, 10, 是], [5, 12, 9, 9, 是], [5, 13, 8, 8, 是], [5, 14, 7, 7, 是], [5, 15, 6, 6, 是], [5, 16, 5, 5, 是], [5, 17, 4, 4, 是], [5, 18, 3, 3, 是], [5, 19, 2, 2, 是], [5, 20, 1, 1, 是], [5, 21, 0, 0, 是], [5, 22, 25, 25, 是], [5, 23, 24, 24, 是], [5, 24, 23, 23, 是], [5, 25, 22, 22, 是], [6, 0, 20, 20, 是], [6, 1, 19, 19, 是], [6, 2, 18, 18, 是], [6, 3, 17, 17, 是], [6, 4, 16, 16, 是], [6, 5, 15, 15, 是], [6, 6, 14, 14, 是], [6, 7, 13, 13, 是], [6, 8, 12, 12, 是], [6, 9, 11, 11, 是], [6, 10, 10, 10, 是], [6, 11, 9, 9, 是], [6, 12, 8, 8, 是], [6, 13, 7, 7, 是], [6, 14, 6, 6, 是], [6, 15, 5, 5, 是], [6, 16, 4, 4, 是], [6, 17, 3, 3, 是], [6, 18, 2, 2, 是], [6, 19, 1, 1, 是], [6, 20, 0, 0, 是], [6, 21, 25, 25, 是], [6, 22, 24, 24, 是], [6, 23, 23, 23, 是], [6, 24, 22, 22, 是], [6, 25, 21, 21, 是], [7, 0, 19, 19, 是], [7, 1, 18, 18, 是], [7, 2, 17, 17, 是], [7, 3, 16, 16, 是], [7, 4, 15, 15, 是], [7, 5, 14, 14, 是], [7, 6, 13, 13, 是], [7, 7, 12, 12, 是], [7, 8, 11, 11, 是], [7, 9, 10, 10, 是], [7, 10, 9, 9, 是], [7, 11, 8, 8, 是], [7, 12, 7, 7, 是], [7, 13, 6, 6, 是], [7, 14, 5, 5, 是], [7, 15, 4, 4, 是], [7, 16, 3, 3, 是], [7, 17, 2, 2, 是], [7, 18, 1, 1, 是], [7, 19, 0, 0, 是], [7, 20, 25, 25, 是], [7, 21, 24, 24, 是], [7, 22, 23, 23, 是], [7, 23, 22, 22, 是], [7, 24, 21, 21, 是], [7, 25, 20, 20, 是], [8, 0, 18, 18, 是], [8, 1, 17, 17, 是], [8, 2, 16, 16, 是], [8, 3, 15, 15, 是], [8, 4, 14, 14, 是], [8, 5, 13, 13, 是], [8, 6, 12, 12, 是], [8, 7, 11, 11, 是], [8, 8, 10, 10, 是], [8, 9, 9, 9, 是], [8, 10, 8, 8, 是], [8, 11, 7, 7, 是], [8, 12, 6, 6, 是], [8, 13, 5, 5, 是], [8, 14, 4, 4, 是], [8, 15, 3, 3, 是], [8, 16, 2, 2, 是], [8, 17, 1, 1, 是], [8, 18, 0, 0, 是], [8, 19, 25, 25, 是], [8, 20, 24, 24, 是], [8, 21, 23, 23, 是], [8, 22, 22, 22, 是], [8, 23, 21, 21, 是], [8, 24, 20, 20, 是], [8, 25, 19, 19, 是], [9, 0, 17, 17, 是], [9, 1, 16, 16, 是], [9, 2, 15, 15, 是], [9, 3, 14, 14, 是], [9, 4, 13, 13, 是], [9, 5, 12, 12, 是], [9, 6, 11, 11, 是], [9, 7, 10, 10, 是], [9, 8, 9, 9, 是], [9, 9, 8, 8, 是], [9, 10, 7, 7, 是], [9, 11, 6, 6, 是], [9, 12, 5, 5, 是], [9, 13, 4, 4, 是], [9, 14, 3, 3, 是], [9, 15, 2, 2, 是], [9, 16, 1, 1, 是], [9, 17, 0, 0, 是], [9, 18, 25, 25, 是], [9, 19, 24, 24, 是], [9, 20, 23, 23, 是], [9, 21, 22, 22, 是], [9, 22, 21, 21, 是], [9, 23, 20, 20, 是], [9, 24, 19, 19, 是], [9, 25, 18, 18, 是], [10, 0, 16, 16, 是], [10, 1, 15, 15, 是], [10, 2, 14, 14, 是], [10, 3, 13, 13, 是], [10, 4, 12, 12, 是], [10, 5, 11, 11, 是], [10, 6, 10, 10, 是], [10, 7, 9, 9, 是], [10, 8, 8, 8, 是], [10, 9, 7, 7, 是], [10, 10, 6, 6, 是], [10, 11, 5, 5, 是], [10, 12, 4, 4, 是], [10, 13, 3, 3, 是], [10, 14, 2, 2, 是], [10, 15, 1, 1, 是], [10, 16, 0, 0, 是], [10, 17, 25, 25, 是], [10, 18, 24, 24, 是], [10, 19, 23, 23, 是], [10, 20, 22, 22, 是], [10, 21, 21, 21, 是], [10, 22, 20, 20, 是], [10, 23, 19, 19, 是], [10, 24, 18, 18, 是], [10, 25, 17, 17, 是], [11, 0, 15, 15, 是], [11, 1, 14, 14, 是], [11, 2, 13, 13, 是], [11, 3, 12, 12, 是], [11, 4, 11, 11, 是], [11, 5, 10, 10, 是], [11, 6, 9, 9, 是], [11, 7, 8, 8, 是], [11, 8, 7, 7, 是], [11, 9, 6, 6, 是], [11, 10, 5, 5, 是], [11, 11, 4, 4, 是], [11, 12, 3, 3, 是], [11, 13, 2, 2, 是], [11, 14, 1, 1, 是], [11, 15, 0, 0, 是], [11, 16, 25, 25, 是], [11, 17, 24, 24, 是], [11, 18, 23, 23, 是], [11, 19, 22, 22, 是], [11, 20, 21, 21, 是], [11, 21, 20, 20, 是], [11, 22, 19, 19, 是], [11, 23, 18, 18, 是], [11, 24, 17, 17, 是], [11, 25, 16, 16, 是], [12, 0, 14, 14, 是], [12, 1, 13, 13, 是], [12, 2, 12, 12, 是], [12, 3, 11, 11, 是], [12, 4, 10, 10, 是], [12, 5, 9, 9, 是], [12, 6, 8, 8, 是], [12, 7, 7, 7, 是], [12, 8, 6, 6, 是], [12, 9, 5, 5, 是], [12, 10, 4, 4, 是], [12, 11, 3, 3, 是], [12, 12, 2, 2, 是], [12, 13, 1, 1, 是], [12, 14, 0, 0, 是], [12, 15, 25, 25, 是], [12, 16, 24, 24, 是], [12, 17, 23, 23, 是], [12, 18, 22, 22, 是], [12, 19, 21, 21, 是], [12, 20, 20, 20, 是], [12, 21, 19, 19, 是], [12, 22, 18, 18, 是], [12, 23, 17, 17, 是], [12, 24, 16, 16, 是], [12, 25, 15, 15, 是], [13, 0, 13, 13, 是], [13, 1, 12, 12, 是], [13, 2, 11, 11, 是], [13, 3, 10, 10, 是], [13, 4, 9, 9, 是], [13, 5, 8, 8, 是], [13, 6, 7, 7, 是], [13, 7, 6, 6, 是], [13, 8, 5, 5, 是], [13, 9, 4, 4, 是], [13, 10, 3, 3, 是], [13, 11, 2, 2, 是], [13, 12, 1, 1, 是], [13, 13, 0, 0, 是], [13, 14, 25, 25, 是], [13, 15, 24, 24, 是], [13, 16, 23, 23, 是], [13, 17, 22, 22, 是], [13, 18, 21, 21, 是], [13, 19, 20, 20, 是], [13, 20, 19, 19, 是], [13, 21, 18, 18, 是], [13, 22, 17, 17, 是], [13, 23, 16, 16, 是], [13, 24, 15, 15, 是], [13, 25, 14, 14, 是], [14, 0, 12, 12, 是], [14, 1, 11, 11, 是], [14, 2, 10, 10, 是], [14, 3, 9, 9, 是], [14, 4, 8, 8, 是], [14, 5, 7, 7, 是], [14, 6, 6, 6, 是], [14, 7, 5, 5, 是], [14, 8, 4, 4, 是], [14, 9, 3, 3, 是], [14, 10, 2, 2, 是], [14, 11, 1, 1, 是], [14, 12, 0, 0, 是], [14, 13, 25, 25, 是], [14, 14, 24, 24, 是], [14, 15, 23, 23, 是], [14, 16, 22, 22, 是], [14, 17, 21, 21, 是], [14, 18, 20, 20, 是], [14, 19, 19, 19, 是], [14, 20, 18, 18, 是], [14, 21, 17, 17, 是], [14, 22, 16, 16, 是], [14, 23, 15, 15, 是], [14, 24, 14, 14, 是], [14, 25, 13, 13, 是], [15, 0, 11, 11, 是], [15, 1, 10, 10, 是], [15, 2, 9, 9, 是], [15, 3, 8, 8, 是], [15, 4, 7, 7, 是], [15, 5, 6, 6, 是], [15, 6, 5, 5, 是], [15, 7, 4, 4, 是], [15, 8, 3, 3, 是], [15, 9, 2, 2, 是], [15, 10, 1, 1, 是], [15, 11, 0, 0, 是], [15, 12, 25, 25, 是], [15, 13, 24, 24, 是], [15, 14, 23, 23, 是], [15, 15, 22, 22, 是], [15, 16, 21, 21, 是], [15, 17, 20, 20, 是], [15, 18, 19, 19, 是], [15, 19, 18, 18, 是], [15, 20, 17, 17, 是], [15, 21, 16, 16, 是], [15, 22, 15, 15, 是], [15, 23, 14, 14, 是], [15, 24, 13, 13, 是], [15, 25, 12, 12, 是], [16, 0, 10, 10, 是], [16, 1, 9, 9, 是], [16, 2, 8, 8, 是], [16, 3, 7, 7, 是], [16, 4, 6, 6, 是], [16, 5, 5, 5, 是], [16, 6, 4, 4, 是], [16, 7, 3, 3, 是], [16, 8, 2, 2, 是], [16, 9, 1, 1, 是], [16, 10, 0, 0, 是], [16, 11, 25, 25, 是], [16, 12, 24, 24, 是], [16, 13, 23, 23, 是], [16, 14, 22, 22, 是], [16, 15, 21, 21, 是], [16, 16, 20, 20, 是], [16, 17, 19, 19, 是], [16, 18, 18, 18, 是], [16, 19, 17, 17, 是], [16, 20, 16, 16, 是], [16, 21, 15, 15, 是], [16, 22, 14, 14, 是], [16, 23, 13, 13, 是], [16, 24, 12, 12, 是], [16, 25, 11, 11, 是], [17, 0, 9, 9, 是], [17, 1, 8, 8, 是], [17, 2, 7, 7, 是], [17, 3, 6, 6, 是], [17, 4, 5, 5, 是], [17, 5, 4, 4, 是], [17, 6, 3, 3, 是], [17, 7, 2, 2, 是], [17, 8, 1, 1, 是], [17, 9, 0, 0, 是], [17, 10, 25, 25, 是], [17, 11, 24, 24, 是], [17, 12, 23, 23, 是], [17, 13, 22, 22, 是], [17, 14, 21, 21, 是], [17, 15, 20, 20, 是], [17, 16, 19, 19, 是], [17, 17, 18, 18, 是], [17, 18, 17, 17, 是], [17, 19, 16, 16, 是], [17, 20, 15, 15, 是], [17, 21, 14, 14, 是], [17, 22, 13, 13, 是], [17, 23, 12, 12, 是], [17, 24, 11, 11, 是], [17, 25, 10, 10, 是], [18, 0, 8, 8, 是], [18, 1, 7, 7, 是], [18, 2, 6, 6, 是], [18, 3, 5, 5, 是], [18, 4, 4, 4, 是], [18, 5, 3, 3, 是], [18, 6, 2, 2, 是], [18, 7, 1, 1, 是], [18, 8, 0, 0, 是], [18, 9, 25, 25, 是], [18, 10, 24, 24, 是], [18, 11, 23, 23, 是], [18, 12, 22, 22, 是], [18, 13, 21, 21, 是], [18, 14, 20, 20, 是], [18, 15, 19, 19, 是], [18, 16, 18, 18, 是], [18, 17, 17, 17, 是], [18, 18, 16, 16, 是], [18, 19, 15, 15, 是], [18, 20, 14, 14, 是], [18, 21, 13, 13, 是], [18, 22, 12, 12, 是], [18, 23, 11, 11, 是], [18, 24, 10, 10, 是], [18, 25, 9, 9, 是], [19, 0, 7, 7, 是], [19, 1, 6, 6, 是], [19, 2, 5, 5, 是], [19, 3, 4, 4, 是], [19, 4, 3, 3, 是], [19, 5, 2, 2, 是], [19, 6, 1, 1, 是], [19, 7, 0, 0, 是], [19, 8, 25, 25, 是], [19, 9, 24, 24, 是], [19, 10, 23, 23, 是], [19, 11, 22, 22, 是], [19, 12, 21, 21, 是], [19, 13, 20, 20, 是], [19, 14, 19, 19, 是], [19, 15, 18, 18, 是], [19, 16, 17, 17, 是], [19, 17, 16, 16, 是], [19, 18, 15, 15, 是], [19, 19, 14, 14, 是], [19, 20, 13, 13, 是], [19, 21, 12, 12, 是], [19, 22, 11, 11, 是], [19, 23, 10, 10, 是], [19, 24, 9, 9, 是], [19, 25, 8, 8, 是], [20, 0, 6, 6, 是], [20, 1, 5, 5, 是], [20, 2, 4, 4, 是], [20, 3, 3, 3, 是], [20, 4, 2, 2, 是], [20, 5, 1, 1, 是], [20, 6, 0, 0, 是], [20, 7, 25, 25, 是], [20, 8, 24, 24, 是], [20, 9, 23, 23, 是], [20, 10, 22, 22, 是], [20, 11, 21, 21, 是], [20, 12, 20, 20, 是], [20, 13, 19, 19, 是], [20, 14, 18, 18, 是], [20, 15, 17, 17, 是], [20, 16, 16, 16, 是], [20, 17, 15, 15, 是], [20, 18, 14, 14, 是], [20, 19, 13, 13, 是], [20, 20, 12, 12, 是], [20, 21, 11, 11, 是], [20, 22, 10, 10, 是], [20, 23, 9, 9, 是], [20, 24, 8, 8, 是], [20, 25, 7, 7, 是], [21, 0, 5, 5, 是], [21, 1, 4, 4, 是], [21, 2, 3, 3, 是], [21, 3, 2, 2, 是], [21, 4, 1, 1, 是], [21, 5, 0, 0, 是], [21, 6, 25, 25, 是], [21, 7, 24, 24, 是], [21, 8, 23, 23, 是], [21, 9, 22, 22, 是], [21, 10, 21, 21, 是], [21, 11, 20, 20, 是], [21, 12, 19, 19, 是], [21, 13, 18, 18, 是], [21, 14, 17, 17, 是], [21, 15, 16, 16, 是], [21, 16, 15, 15, 是], [21, 17, 14, 14, 是], [21, 18, 13, 13, 是], [21, 19, 12, 12, 是], [21, 20, 11, 11, 是], [21, 21, 10, 10, 是], [21, 22, 9, 9, 是], [21, 23, 8, 8, 是], [21, 24, 7, 7, 是], [21, 25, 6, 6, 是], [22, 0, 4, 4, 是], [22, 1, 3, 3, 是], [22, 2, 2, 2, 是], [22, 3, 1, 1, 是], [22, 4, 0, 0, 是], [22, 5, 25, 25, 是], [22, 6, 24, 24, 是], [22, 7, 23, 23, 是], [22, 8, 22, 22, 是], [22, 9, 21, 21, 是], [22, 10, 20, 20, 是], [22, 11, 19, 19, 是], [22, 12, 18, 18, 是], [22, 13, 17, 17, 是], [22, 14, 16, 16, 是], [22, 15, 15, 15, 是], [22, 16, 14, 14, 是], [22, 17, 13, 13, 是], [22, 18, 12, 12, 是], [22, 19, 11, 11, 是], [22, 20, 10, 10, 是], [22, 21, 9, 9, 是], [22, 22, 8, 8, 是], [22, 23, 7, 7, 是], [22, 24, 6, 6, 是], [22, 25, 5, 5, 是], [23, 0, 3, 3, 是], [23, 1, 2, 2, 是], [23, 2, 1, 1, 是], [23, 3, 0, 0, 是], [23, 4, 25, 25, 是], [23, 5, 24, 24, 是], [23, 6, 23, 23, 是], [23, 7, 22, 22, 是], [23, 8, 21, 21, 是], [23, 9, 20, 20, 是], [23, 10, 19, 19, 是], [23, 11, 18, 18, 是], [23, 12, 17, 17, 是], [23, 13, 16, 16, 是], [23, 14, 15, 15, 是], [23, 15, 14, 14, 是], [23, 16, 13, 13, 是], [23, 17, 12, 12, 是], [23, 18, 11, 11, 是], [23, 19, 10, 10, 是], [23, 20, 9, 9, 是], [23, 21, 8, 8, 是], [23, 22, 7, 7, 是], [23, 23, 6, 6, 是], [23, 24, 5, 5, 是], [23, 25, 4, 4, 是], [24, 0, 2, 2, 是], [24, 1, 1, 1, 是], [24, 2, 0, 0, 是], [24, 3, 25, 25, 是], [24, 4, 24, 24, 是], [24, 5, 23, 23, 是], [24, 6, 22, 22, 是], [24, 7, 21, 21, 是], [24, 8, 20, 20, 是], [24, 9, 19, 19, 是], [24, 10, 18, 18, 是], [24, 11, 17, 17, 是], [24, 12, 16, 16, 是], [24, 13, 15, 15, 是], [24, 14, 14, 14, 是], [24, 15, 13, 13, 是], [24, 16, 12, 12, 是], [24, 17, 11, 11, 是], [24, 18, 10, 10, 是], [24, 19, 9, 9, 是], [24, 20, 8, 8, 是], [24, 21, 7, 7, 是], [24, 22, 6, 6, 是], [24, 23, 5, 5, 是], [24, 24, 4, 4, 是], [24, 25, 3, 3, 是], [25, 0, 1, 1, 是], [25, 1, 0, 0, 是], [25, 2, 25, 25, 是], [25, 3, 24, 24, 是], [25, 4, 23, 23, 是], [25, 5, 22, 22, 是], [25, 6, 21, 21, 是], [25, 7, 20, 20, 是], [25, 8, 19, 19, 是], [25, 9, 18, 18, 是], [25, 10, 17, 17, 是], [25, 11, 16, 16, 是], [25, 12, 15, 15, 是], [25, 13, 14, 14, 是], [25, 14, 13, 13, 是], [25, 15, 12, 12, 是], [25, 16, 11, 11, 是], [25, 17, 10, 10, 是], [25, 18, 9, 9, 是], [25, 19, 8, 8, 是], [25, 20, 7, 7, 是], [25, 21, 6, 6, 是], [25, 22, 5, 5, 是], [25, 23, 4, 4, 是], [25, 24, 3, 3, 是], [25, 25, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(26, 26, [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 25, 25, 是], [0, 2, 24, 24, 是], [0, 3, 23, 23, 是], [0, 4, 22, 22, 是], [0, 5, 21, 21, 是], [0, 6, 20, 20, 是], [0, 7, 19, 19, 是], [0, 8, 18, 18, 是], [0, 9, 17, 17, 是], [0, 10, 16, 16, 是], [0, 11, 15, 15, 是], [0, 12, 14, 14, 是], [0, 13, 13, 13, 是], [0, 14, 12, 12, 是], [0, 15, 11, 11, 是], [0, 16, 10, 10, 是], [0, 17, 9, 9, 是], [0, 18, 8, 8, 是], [0, 19, 7, 7, 是], [0, 20, 6, 6, 是], [0, 21, 5, 5, 是], [0, 22, 4, 4, 是], [0, 23, 3, 3, 是], [0, 24, 2, 2, 是], [0, 25, 1, 1, 是], [1, 0, 25, 25, 是], [1, 1, 24, 24, 是], [1, 2, 23, 23, 是], [1, 3, 22, 22, 是], [1, 4, 21, 21, 是], [1, 5, 20, 20, 是], [1, 6, 19, 19, 是], [1, 7, 18, 18, 是], [1, 8, 17, 17, 是], [1, 9, 16, 16, 是], [1, 10, 15, 15, 是], [1, 11, 14, 14, 是], [1, 12, 13, 13, 是], [1, 13, 12, 12, 是], [1, 14, 11, 11, 是], [1, 15, 10, 10, 是], [1, 16, 9, 9, 是], [1, 17, 8, 8, 是], [1, 18, 7, 7, 是], [1, 19, 6, 6, 是], [1, 20, 5, 5, 是], [1, 21, 4, 4, 是], [1, 22, 3, 3, 是], [1, 23, 2, 2, 是], [1, 24, 1, 1, 是], [1, 25, 0, 0, 是], [2, 0, 24, 24, 是], [2, 1, 23, 23, 是], [2, 2, 22, 22, 是], [2, 3, 21, 21, 是], [2, 4, 20, 20, 是], [2, 5, 19, 19, 是], [2, 6, 18, 18, 是], [2, 7, 17, 17, 是], [2, 8, 16, 16, 是], [2, 9, 15, 15, 是], [2, 10, 14, 14, 是], [2, 11, 13, 13, 是], [2, 12, 12, 12, 是], [2, 13, 11, 11, 是], [2, 14, 10, 10, 是], [2, 15, 9, 9, 是], [2, 16, 8, 8, 是], [2, 17, 7, 7, 是], [2, 18, 6, 6, 是], [2, 19, 5, 5, 是], [2, 20, 4, 4, 是], [2, 21, 3, 3, 是], [2, 22, 2, 2, 是], [2, 23, 1, 1, 是], [2, 24, 0, 0, 是], [2, 25, 25, 25, 是], [3, 0, 23, 23, 是], [3, 1, 22, 22, 是], [3, 2, 21, 21, 是], [3, 3, 20, 20, 是], [3, 4, 19, 19, 是], [3, 5, 18, 18, 是], [3, 6, 17, 17, 是], [3, 7, 16, 16, 是], [3, 8, 15, 15, 是], [3, 9, 14, 14, 是], [3, 10, 13, 13, 是], [3, 11, 12, 12, 是], [3, 12, 11, 11, 是], [3, 13, 10, 10, 是], [3, 14, 9, 9, 是], [3, 15, 8, 8, 是], [3, 16, 7, 7, 是], [3, 17, 6, 6, 是], [3, 18, 5, 5, 是], [3, 19, 4, 4, 是], [3, 20, 3, 3, 是], [3, 21, 2, 2, 是], [3, 22, 1, 1, 是], [3, 23, 0, 0, 是], [3, 24, 25, 25, 是], [3, 25, 24, 24, 是], [4, 0, 22, 22, 是], [4, 1, 21, 21, 是], [4, 2, 20, 20, 是], [4, 3, 19, 19, 是], [4, 4, 18, 18, 是], [4, 5, 17, 17, 是], [4, 6, 16, 16, 是], [4, 7, 15, 15, 是], [4, 8, 14, 14, 是], [4, 9, 13, 13, 是], [4, 10, 12, 12, 是], [4, 11, 11, 11, 是], [4, 12, 10, 10, 是], [4, 13, 9, 9, 是], [4, 14, 8, 8, 是], [4, 15, 7, 7, 是], [4, 16, 6, 6, 是], [4, 17, 5, 5, 是], [4, 18, 4, 4, 是], [4, 19, 3, 3, 是], [4, 20, 2, 2, 是], [4, 21, 1, 1, 是], [4, 22, 0, 0, 是], [4, 23, 25, 25, 是], [4, 24, 24, 24, 是], [4, 25, 23, 23, 是], [5, 0, 21, 21, 是], [5, 1, 20, 20, 是], [5, 2, 19, 19, 是], [5, 3, 18, 18, 是], [5, 4, 17, 17, 是], [5, 5, 16, 16, 是], [5, 6, 15, 15, 是], [5, 7, 14, 14, 是], [5, 8, 13, 13, 是], [5, 9, 12, 12, 是], [5, 10, 11, 11, 是], [5, 11, 10, 10, 是], [5, 12, 9, 9, 是], [5, 13, 8, 8, 是], [5, 14, 7, 7, 是], [5, 15, 6, 6, 是], [5, 16, 5, 5, 是], [5, 17, 4, 4, 是], [5, 18, 3, 3, 是], [5, 19, 2, 2, 是], [5, 20, 1, 1, 是], [5, 21, 0, 0, 是], [5, 22, 25, 25, 是], [5, 23, 24, 24, 是], [5, 24, 23, 23, 是], [5, 25, 22, 22, 是], [6, 0, 20, 20, 是], [6, 1, 19, 19, 是], [6, 2, 18, 18, 是], [6, 3, 17, 17, 是], [6, 4, 16, 16, 是], [6, 5, 15, 15, 是], [6, 6, 14, 14, 是], [6, 7, 13, 13, 是], [6, 8, 12, 12, 是], [6, 9, 11, 11, 是], [6, 10, 10, 10, 是], [6, 11, 9, 9, 是], [6, 12, 8, 8, 是], [6, 13, 7, 7, 是], [6, 14, 6, 6, 是], [6, 15, 5, 5, 是], [6, 16, 4, 4, 是], [6, 17, 3, 3, 是], [6, 18, 2, 2, 是], [6, 19, 1, 1, 是], [6, 20, 0, 0, 是], [6, 21, 25, 25, 是], [6, 22, 24, 24, 是], [6, 23, 23, 23, 是], [6, 24, 22, 22, 是], [6, 25, 21, 21, 是], [7, 0, 19, 19, 是], [7, 1, 18, 18, 是], [7, 2, 17, 17, 是], [7, 3, 16, 16, 是], [7, 4, 15, 15, 是], [7, 5, 14, 14, 是], [7, 6, 13, 13, 是], [7, 7, 12, 12, 是], [7, 8, 11, 11, 是], [7, 9, 10, 10, 是], [7, 10, 9, 9, 是], [7, 11, 8, 8, 是], [7, 12, 7, 7, 是], [7, 13, 6, 6, 是], [7, 14, 5, 5, 是], [7, 15, 4, 4, 是], [7, 16, 3, 3, 是], [7, 17, 2, 2, 是], [7, 18, 1, 1, 是], [7, 19, 0, 0, 是], [7, 20, 25, 25, 是], [7, 21, 24, 24, 是], [7, 22, 23, 23, 是], [7, 23, 22, 22, 是], [7, 24, 21, 21, 是], [7, 25, 20, 20, 是], [8, 0, 18, 18, 是], [8, 1, 17, 17, 是], [8, 2, 16, 16, 是], [8, 3, 15, 15, 是], [8, 4, 14, 14, 是], [8, 5, 13, 13, 是], [8, 6, 12, 12, 是], [8, 7, 11, 11, 是], [8, 8, 10, 10, 是], [8, 9, 9, 9, 是], [8, 10, 8, 8, 是], [8, 11, 7, 7, 是], [8, 12, 6, 6, 是], [8, 13, 5, 5, 是], [8, 14, 4, 4, 是], [8, 15, 3, 3, 是], [8, 16, 2, 2, 是], [8, 17, 1, 1, 是], [8, 18, 0, 0, 是], [8, 19, 25, 25, 是], [8, 20, 24, 24, 是], [8, 21, 23, 23, 是], [8, 22, 22, 22, 是], [8, 23, 21, 21, 是], [8, 24, 20, 20, 是], [8, 25, 19, 19, 是], [9, 0, 17, 17, 是], [9, 1, 16, 16, 是], [9, 2, 15, 15, 是], [9, 3, 14, 14, 是], [9, 4, 13, 13, 是], [9, 5, 12, 12, 是], [9, 6, 11, 11, 是], [9, 7, 10, 10, 是], [9, 8, 9, 9, 是], [9, 9, 8, 8, 是], [9, 10, 7, 7, 是], [9, 11, 6, 6, 是], [9, 12, 5, 5, 是], [9, 13, 4, 4, 是], [9, 14, 3, 3, 是], [9, 15, 2, 2, 是], [9, 16, 1, 1, 是], [9, 17, 0, 0, 是], [9, 18, 25, 25, 是], [9, 19, 24, 24, 是], [9, 20, 23, 23, 是], [9, 21, 22, 22, 是], [9, 22, 21, 21, 是], [9, 23, 20, 20, 是], [9, 24, 19, 19, 是], [9, 25, 18, 18, 是], [10, 0, 16, 16, 是], [10, 1, 15, 15, 是], [10, 2, 14, 14, 是], [10, 3, 13, 13, 是], [10, 4, 12, 12, 是], [10, 5, 11, 11, 是], [10, 6, 10, 10, 是], [10, 7, 9, 9, 是], [10, 8, 8, 8, 是], [10, 9, 7, 7, 是], [10, 10, 6, 6, 是], [10, 11, 5, 5, 是], [10, 12, 4, 4, 是], [10, 13, 3, 3, 是], [10, 14, 2, 2, 是], [10, 15, 1, 1, 是], [10, 16, 0, 0, 是], [10, 17, 25, 25, 是], [10, 18, 24, 24, 是], [10, 19, 23, 23, 是], [10, 20, 22, 22, 是], [10, 21, 21, 21, 是], [10, 22, 20, 20, 是], [10, 23, 19, 19, 是], [10, 24, 18, 18, 是], [10, 25, 17, 17, 是], [11, 0, 15, 15, 是], [11, 1, 14, 14, 是], [11, 2, 13, 13, 是], [11, 3, 12, 12, 是], [11, 4, 11, 11, 是], [11, 5, 10, 10, 是], [11, 6, 9, 9, 是], [11, 7, 8, 8, 是], [11, 8, 7, 7, 是], [11, 9, 6, 6, 是], [11, 10, 5, 5, 是], [11, 11, 4, 4, 是], [11, 12, 3, 3, 是], [11, 13, 2, 2, 是], [11, 14, 1, 1, 是], [11, 15, 0, 0, 是], [11, 16, 25, 25, 是], [11, 17, 24, 24, 是], [11, 18, 23, 23, 是], [11, 19, 22, 22, 是], [11, 20, 21, 21, 是], [11, 21, 20, 20, 是], [11, 22, 19, 19, 是], [11, 23, 18, 18, 是], [11, 24, 17, 17, 是], [11, 25, 16, 16, 是], [12, 0, 14, 14, 是], [12, 1, 13, 13, 是], [12, 2, 12, 12, 是], [12, 3, 11, 11, 是], [12, 4, 10, 10, 是], [12, 5, 9, 9, 是], [12, 6, 8, 8, 是], [12, 7, 7, 7, 是], [12, 8, 6, 6, 是], [12, 9, 5, 5, 是], [12, 10, 4, 4, 是], [12, 11, 3, 3, 是], [12, 12, 2, 2, 是], [12, 13, 1, 1, 是], [12, 14, 0, 0, 是], [12, 15, 25, 25, 是], [12, 16, 24, 24, 是], [12, 17, 23, 23, 是], [12, 18, 22, 22, 是], [12, 19, 21, 21, 是], [12, 20, 20, 20, 是], [12, 21, 19, 19, 是], [12, 22, 18, 18, 是], [12, 23, 17, 17, 是], [12, 24, 16, 16, 是], [12, 25, 15, 15, 是], [13, 0, 13, 13, 是], [13, 1, 12, 12, 是], [13, 2, 11, 11, 是], [13, 3, 10, 10, 是], [13, 4, 9, 9, 是], [13, 5, 8, 8, 是], [13, 6, 7, 7, 是], [13, 7, 6, 6, 是], [13, 8, 5, 5, 是], [13, 9, 4, 4, 是], [13, 10, 3, 3, 是], [13, 11, 2, 2, 是], [13, 12, 1, 1, 是], [13, 13, 0, 0, 是], [13, 14, 25, 25, 是], [13, 15, 24, 24, 是], [13, 16, 23, 23, 是], [13, 17, 22, 22, 是], [13, 18, 21, 21, 是], [13, 19, 20, 20, 是], [13, 20, 19, 19, 是], [13, 21, 18, 18, 是], [13, 22, 17, 17, 是], [13, 23, 16, 16, 是], [13, 24, 15, 15, 是], [13, 25, 14, 14, 是], [14, 0, 12, 12, 是], [14, 1, 11, 11, 是], [14, 2, 10, 10, 是], [14, 3, 9, 9, 是], [14, 4, 8, 8, 是], [14, 5, 7, 7, 是], [14, 6, 6, 6, 是], [14, 7, 5, 5, 是], [14, 8, 4, 4, 是], [14, 9, 3, 3, 是], [14, 10, 2, 2, 是], [14, 11, 1, 1, 是], [14, 12, 0, 0, 是], [14, 13, 25, 25, 是], [14, 14, 24, 24, 是], [14, 15, 23, 23, 是], [14, 16, 22, 22, 是], [14, 17, 21, 21, 是], [14, 18, 20, 20, 是], [14, 19, 19, 19, 是], [14, 20, 18, 18, 是], [14, 21, 17, 17, 是], [14, 22, 16, 16, 是], [14, 23, 15, 15, 是], [14, 24, 14, 14, 是], [14, 25, 13, 13, 是], [15, 0, 11, 11, 是], [15, 1, 10, 10, 是], [15, 2, 9, 9, 是], [15, 3, 8, 8, 是], [15, 4, 7, 7, 是], [15, 5, 6, 6, 是], [15, 6, 5, 5, 是], [15, 7, 4, 4, 是], [15, 8, 3, 3, 是], [15, 9, 2, 2, 是], [15, 10, 1, 1, 是], [15, 11, 0, 0, 是], [15, 12, 25, 25, 是], [15, 13, 24, 24, 是], [15, 14, 23, 23, 是], [15, 15, 22, 22, 是], [15, 16, 21, 21, 是], [15, 17, 20, 20, 是], [15, 18, 19, 19, 是], [15, 19, 18, 18, 是], [15, 20, 17, 17, 是], [15, 21, 16, 16, 是], [15, 22, 15, 15, 是], [15, 23, 14, 14, 是], [15, 24, 13, 13, 是], [15, 25, 12, 12, 是], [16, 0, 10, 10, 是], [16, 1, 9, 9, 是], [16, 2, 8, 8, 是], [16, 3, 7, 7, 是], [16, 4, 6, 6, 是], [16, 5, 5, 5, 是], [16, 6, 4, 4, 是], [16, 7, 3, 3, 是], [16, 8, 2, 2, 是], [16, 9, 1, 1, 是], [16, 10, 0, 0, 是], [16, 11, 25, 25, 是], [16, 12, 24, 24, 是], [16, 13, 23, 23, 是], [16, 14, 22, 22, 是], [16, 15, 21, 21, 是], [16, 16, 20, 20, 是], [16, 17, 19, 19, 是], [16, 18, 18, 18, 是], [16, 19, 17, 17, 是], [16, 20, 16, 16, 是], [16, 21, 15, 15, 是], [16, 22, 14, 14, 是], [16, 23, 13, 13, 是], [16, 24, 12, 12, 是], [16, 25, 11, 11, 是], [17, 0, 9, 9, 是], [17, 1, 8, 8, 是], [17, 2, 7, 7, 是], [17, 3, 6, 6, 是], [17, 4, 5, 5, 是], [17, 5, 4, 4, 是], [17, 6, 3, 3, 是], [17, 7, 2, 2, 是], [17, 8, 1, 1, 是], [17, 9, 0, 0, 是], [17, 10, 25, 25, 是], [17, 11, 24, 24, 是], [17, 12, 23, 23, 是], [17, 13, 22, 22, 是], [17, 14, 21, 21, 是], [17, 15, 20, 20, 是], [17, 16, 19, 19, 是], [17, 17, 18, 18, 是], [17, 18, 17, 17, 是], [17, 19, 16, 16, 是], [17, 20, 15, 15, 是], [17, 21, 14, 14, 是], [17, 22, 13, 13, 是], [17, 23, 12, 12, 是], [17, 24, 11, 11, 是], [17, 25, 10, 10, 是], [18, 0, 8, 8, 是], [18, 1, 7, 7, 是], [18, 2, 6, 6, 是], [18, 3, 5, 5, 是], [18, 4, 4, 4, 是], [18, 5, 3, 3, 是], [18, 6, 2, 2, 是], [18, 7, 1, 1, 是], [18, 8, 0, 0, 是], [18, 9, 25, 25, 是], [18, 10, 24, 24, 是], [18, 11, 23, 23, 是], [18, 12, 22, 22, 是], [18, 13, 21, 21, 是], [18, 14, 20, 20, 是], [18, 15, 19, 19, 是], [18, 16, 18, 18, 是], [18, 17, 17, 17, 是], [18, 18, 16, 16, 是], [18, 19, 15, 15, 是], [18, 20, 14, 14, 是], [18, 21, 13, 13, 是], [18, 22, 12, 12, 是], [18, 23, 11, 11, 是], [18, 24, 10, 10, 是], [18, 25, 9, 9, 是], [19, 0, 7, 7, 是], [19, 1, 6, 6, 是], [19, 2, 5, 5, 是], [19, 3, 4, 4, 是], [19, 4, 3, 3, 是], [19, 5, 2, 2, 是], [19, 6, 1, 1, 是], [19, 7, 0, 0, 是], [19, 8, 25, 25, 是], [19, 9, 24, 24, 是], [19, 10, 23, 23, 是], [19, 11, 22, 22, 是], [19, 12, 21, 21, 是], [19, 13, 20, 20, 是], [19, 14, 19, 19, 是], [19, 15, 18, 18, 是], [19, 16, 17, 17, 是], [19, 17, 16, 16, 是], [19, 18, 15, 15, 是], [19, 19, 14, 14, 是], [19, 20, 13, 13, 是], [19, 21, 12, 12, 是], [19, 22, 11, 11, 是], [19, 23, 10, 10, 是], [19, 24, 9, 9, 是], [19, 25, 8, 8, 是], [20, 0, 6, 6, 是], [20, 1, 5, 5, 是], [20, 2, 4, 4, 是], [20, 3, 3, 3, 是], [20, 4, 2, 2, 是], [20, 5, 1, 1, 是], [20, 6, 0, 0, 是], [20, 7, 25, 25, 是], [20, 8, 24, 24, 是], [20, 9, 23, 23, 是], [20, 10, 22, 22, 是], [20, 11, 21, 21, 是], [20, 12, 20, 20, 是], [20, 13, 19, 19, 是], [20, 14, 18, 18, 是], [20, 15, 17, 17, 是], [20, 16, 16, 16, 是], [20, 17, 15, 15, 是], [20, 18, 14, 14, 是], [20, 19, 13, 13, 是], [20, 20, 12, 12, 是], [20, 21, 11, 11, 是], [20, 22, 10, 10, 是], [20, 23, 9, 9, 是], [20, 24, 8, 8, 是], [20, 25, 7, 7, 是], [21, 0, 5, 5, 是], [21, 1, 4, 4, 是], [21, 2, 3, 3, 是], [21, 3, 2, 2, 是], [21, 4, 1, 1, 是], [21, 5, 0, 0, 是], [21, 6, 25, 25, 是], [21, 7, 24, 24, 是], [21, 8, 23, 23, 是], [21, 9, 22, 22, 是], [21, 10, 21, 21, 是], [21, 11, 20, 20, 是], [21, 12, 19, 19, 是], [21, 13, 18, 18, 是], [21, 14, 17, 17, 是], [21, 15, 16, 16, 是], [21, 16, 15, 15, 是], [21, 17, 14, 14, 是], [21, 18, 13, 13, 是], [21, 19, 12, 12, 是], [21, 20, 11, 11, 是], [21, 21, 10, 10, 是], [21, 22, 9, 9, 是], [21, 23, 8, 8, 是], [21, 24, 7, 7, 是], [21, 25, 6, 6, 是], [22, 0, 4, 4, 是], [22, 1, 3, 3, 是], [22, 2, 2, 2, 是], [22, 3, 1, 1, 是], [22, 4, 0, 0, 是], [22, 5, 25, 25, 是], [22, 6, 24, 24, 是], [22, 7, 23, 23, 是], [22, 8, 22, 22, 是], [22, 9, 21, 21, 是], [22, 10, 20, 20, 是], [22, 11, 19, 19, 是], [22, 12, 18, 18, 是], [22, 13, 17, 17, 是], [22, 14, 16, 16, 是], [22, 15, 15, 15, 是], [22, 16, 14, 14, 是], [22, 17, 13, 13, 是], [22, 18, 12, 12, 是], [22, 19, 11, 11, 是], [22, 20, 10, 10, 是], [22, 21, 9, 9, 是], [22, 22, 8, 8, 是], [22, 23, 7, 7, 是], [22, 24, 6, 6, 是], [22, 25, 5, 5, 是], [23, 0, 3, 3, 是], [23, 1, 2, 2, 是], [23, 2, 1, 1, 是], [23, 3, 0, 0, 是], [23, 4, 25, 25, 是], [23, 5, 24, 24, 是], [23, 6, 23, 23, 是], [23, 7, 22, 22, 是], [23, 8, 21, 21, 是], [23, 9, 20, 20, 是], [23, 10, 19, 19, 是], [23, 11, 18, 18, 是], [23, 12, 17, 17, 是], [23, 13, 16, 16, 是], [23, 14, 15, 15, 是], [23, 15, 14, 14, 是], [23, 16, 13, 13, 是], [23, 17, 12, 12, 是], [23, 18, 11, 11, 是], [23, 19, 10, 10, 是], [23, 20, 9, 9, 是], [23, 21, 8, 8, 是], [23, 22, 7, 7, 是], [23, 23, 6, 6, 是], [23, 24, 5, 5, 是], [23, 25, 4, 4, 是], [24, 0, 2, 2, 是], [24, 1, 1, 1, 是], [24, 2, 0, 0, 是], [24, 3, 25, 25, 是], [24, 4, 24, 24, 是], [24, 5, 23, 23, 是], [24, 6, 22, 22, 是], [24, 7, 21, 21, 是], [24, 8, 20, 20, 是], [24, 9, 19, 19, 是], [24, 10, 18, 18, 是], [24, 11, 17, 17, 是], [24, 12, 16, 16, 是], [24, 13, 15, 15, 是], [24, 14, 14, 14, 是], [24, 15, 13, 13, 是], [24, 16, 12, 12, 是], [24, 17, 11, 11, 是], [24, 18, 10, 10, 是], [24, 19, 9, 9, 是], [24, 20, 8, 8, 是], [24, 21, 7, 7, 是], [24, 22, 6, 6, 是], [24, 23, 5, 5, 是], [24, 24, 4, 4, 是], [24, 25, 3, 3, 是], [25, 0, 1, 1, 是], [25, 1, 0, 0, 是], [25, 2, 25, 25, 是], [25, 3, 24, 24, 是], [25, 4, 23, 23, 是], [25, 5, 22, 22, 是], [25, 6, 21, 21, 是], [25, 7, 20, 20, 是], [25, 8, 19, 19, 是], [25, 9, 18, 18, 是], [25, 10, 17, 17, 是], [25, 11, 16, 16, 是], [25, 12, 15, 15, 是], [25, 13, 14, 14, 是], [25, 14, 13, 13, 是], [25, 15, 12, 12, 是], [25, 16, 11, 11, 是], [25, 17, 10, 10, 是], [25, 18, 9, 9, 是], [25, 19, 8, 8, 是], [25, 20, 7, 7, 是], [25, 21, 6, 6, 是], [25, 22, 5, 5, 是], [25, 23, 4, 4, 是], [25, 24, 3, 3, 是], [25, 25, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141024,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],"target_identity":0,"mapping":[0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0025","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-homomorphism-qualification；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],\"target_identity\":0,\"mapping\":[0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 27, [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 26, 26, 是], [0, 2, 25, 25, 是], [0, 3, 24, 24, 是], [0, 4, 23, 23, 是], [0, 5, 22, 22, 是], [0, 6, 21, 21, 是], [0, 7, 20, 20, 是], [0, 8, 19, 19, 是], [0, 9, 18, 18, 是], [0, 10, 17, 17, 是], [0, 11, 16, 16, 是], [0, 12, 15, 15, 是], [0, 13, 14, 14, 是], [0, 14, 13, 13, 是], [0, 15, 12, 12, 是], [0, 16, 11, 11, 是], [0, 17, 10, 10, 是], [0, 18, 9, 9, 是], [0, 19, 8, 8, 是], [0, 20, 7, 7, 是], [0, 21, 6, 6, 是], [0, 22, 5, 5, 是], [0, 23, 4, 4, 是], [0, 24, 3, 3, 是], [0, 25, 2, 2, 是], [0, 26, 1, 1, 是], [1, 0, 26, 26, 是], [1, 1, 25, 25, 是], [1, 2, 24, 24, 是], [1, 3, 23, 23, 是], [1, 4, 22, 22, 是], [1, 5, 21, 21, 是], [1, 6, 20, 20, 是], [1, 7, 19, 19, 是], [1, 8, 18, 18, 是], [1, 9, 17, 17, 是], [1, 10, 16, 16, 是], [1, 11, 15, 15, 是], [1, 12, 14, 14, 是], [1, 13, 13, 13, 是], [1, 14, 12, 12, 是], [1, 15, 11, 11, 是], [1, 16, 10, 10, 是], [1, 17, 9, 9, 是], [1, 18, 8, 8, 是], [1, 19, 7, 7, 是], [1, 20, 6, 6, 是], [1, 21, 5, 5, 是], [1, 22, 4, 4, 是], [1, 23, 3, 3, 是], [1, 24, 2, 2, 是], [1, 25, 1, 1, 是], [1, 26, 0, 0, 是], [2, 0, 25, 25, 是], [2, 1, 24, 24, 是], [2, 2, 23, 23, 是], [2, 3, 22, 22, 是], [2, 4, 21, 21, 是], [2, 5, 20, 20, 是], [2, 6, 19, 19, 是], [2, 7, 18, 18, 是], [2, 8, 17, 17, 是], [2, 9, 16, 16, 是], [2, 10, 15, 15, 是], [2, 11, 14, 14, 是], [2, 12, 13, 13, 是], [2, 13, 12, 12, 是], [2, 14, 11, 11, 是], [2, 15, 10, 10, 是], [2, 16, 9, 9, 是], [2, 17, 8, 8, 是], [2, 18, 7, 7, 是], [2, 19, 6, 6, 是], [2, 20, 5, 5, 是], [2, 21, 4, 4, 是], [2, 22, 3, 3, 是], [2, 23, 2, 2, 是], [2, 24, 1, 1, 是], [2, 25, 0, 0, 是], [2, 26, 26, 26, 是], [3, 0, 24, 24, 是], [3, 1, 23, 23, 是], [3, 2, 22, 22, 是], [3, 3, 21, 21, 是], [3, 4, 20, 20, 是], [3, 5, 19, 19, 是], [3, 6, 18, 18, 是], [3, 7, 17, 17, 是], [3, 8, 16, 16, 是], [3, 9, 15, 15, 是], [3, 10, 14, 14, 是], [3, 11, 13, 13, 是], [3, 12, 12, 12, 是], [3, 13, 11, 11, 是], [3, 14, 10, 10, 是], [3, 15, 9, 9, 是], [3, 16, 8, 8, 是], [3, 17, 7, 7, 是], [3, 18, 6, 6, 是], [3, 19, 5, 5, 是], [3, 20, 4, 4, 是], [3, 21, 3, 3, 是], [3, 22, 2, 2, 是], [3, 23, 1, 1, 是], [3, 24, 0, 0, 是], [3, 25, 26, 26, 是], [3, 26, 25, 25, 是], [4, 0, 23, 23, 是], [4, 1, 22, 22, 是], [4, 2, 21, 21, 是], [4, 3, 20, 20, 是], [4, 4, 19, 19, 是], [4, 5, 18, 18, 是], [4, 6, 17, 17, 是], [4, 7, 16, 16, 是], [4, 8, 15, 15, 是], [4, 9, 14, 14, 是], [4, 10, 13, 13, 是], [4, 11, 12, 12, 是], [4, 12, 11, 11, 是], [4, 13, 10, 10, 是], [4, 14, 9, 9, 是], [4, 15, 8, 8, 是], [4, 16, 7, 7, 是], [4, 17, 6, 6, 是], [4, 18, 5, 5, 是], [4, 19, 4, 4, 是], [4, 20, 3, 3, 是], [4, 21, 2, 2, 是], [4, 22, 1, 1, 是], [4, 23, 0, 0, 是], [4, 24, 26, 26, 是], [4, 25, 25, 25, 是], [4, 26, 24, 24, 是], [5, 0, 22, 22, 是], [5, 1, 21, 21, 是], [5, 2, 20, 20, 是], [5, 3, 19, 19, 是], [5, 4, 18, 18, 是], [5, 5, 17, 17, 是], [5, 6, 16, 16, 是], [5, 7, 15, 15, 是], [5, 8, 14, 14, 是], [5, 9, 13, 13, 是], [5, 10, 12, 12, 是], [5, 11, 11, 11, 是], [5, 12, 10, 10, 是], [5, 13, 9, 9, 是], [5, 14, 8, 8, 是], [5, 15, 7, 7, 是], [5, 16, 6, 6, 是], [5, 17, 5, 5, 是], [5, 18, 4, 4, 是], [5, 19, 3, 3, 是], [5, 20, 2, 2, 是], [5, 21, 1, 1, 是], [5, 22, 0, 0, 是], [5, 23, 26, 26, 是], [5, 24, 25, 25, 是], [5, 25, 24, 24, 是], [5, 26, 23, 23, 是], [6, 0, 21, 21, 是], [6, 1, 20, 20, 是], [6, 2, 19, 19, 是], [6, 3, 18, 18, 是], [6, 4, 17, 17, 是], [6, 5, 16, 16, 是], [6, 6, 15, 15, 是], [6, 7, 14, 14, 是], [6, 8, 13, 13, 是], [6, 9, 12, 12, 是], [6, 10, 11, 11, 是], [6, 11, 10, 10, 是], [6, 12, 9, 9, 是], [6, 13, 8, 8, 是], [6, 14, 7, 7, 是], [6, 15, 6, 6, 是], [6, 16, 5, 5, 是], [6, 17, 4, 4, 是], [6, 18, 3, 3, 是], [6, 19, 2, 2, 是], [6, 20, 1, 1, 是], [6, 21, 0, 0, 是], [6, 22, 26, 26, 是], [6, 23, 25, 25, 是], [6, 24, 24, 24, 是], [6, 25, 23, 23, 是], [6, 26, 22, 22, 是], [7, 0, 20, 20, 是], [7, 1, 19, 19, 是], [7, 2, 18, 18, 是], [7, 3, 17, 17, 是], [7, 4, 16, 16, 是], [7, 5, 15, 15, 是], [7, 6, 14, 14, 是], [7, 7, 13, 13, 是], [7, 8, 12, 12, 是], [7, 9, 11, 11, 是], [7, 10, 10, 10, 是], [7, 11, 9, 9, 是], [7, 12, 8, 8, 是], [7, 13, 7, 7, 是], [7, 14, 6, 6, 是], [7, 15, 5, 5, 是], [7, 16, 4, 4, 是], [7, 17, 3, 3, 是], [7, 18, 2, 2, 是], [7, 19, 1, 1, 是], [7, 20, 0, 0, 是], [7, 21, 26, 26, 是], [7, 22, 25, 25, 是], [7, 23, 24, 24, 是], [7, 24, 23, 23, 是], [7, 25, 22, 22, 是], [7, 26, 21, 21, 是], [8, 0, 19, 19, 是], [8, 1, 18, 18, 是], [8, 2, 17, 17, 是], [8, 3, 16, 16, 是], [8, 4, 15, 15, 是], [8, 5, 14, 14, 是], [8, 6, 13, 13, 是], [8, 7, 12, 12, 是], [8, 8, 11, 11, 是], [8, 9, 10, 10, 是], [8, 10, 9, 9, 是], [8, 11, 8, 8, 是], [8, 12, 7, 7, 是], [8, 13, 6, 6, 是], [8, 14, 5, 5, 是], [8, 15, 4, 4, 是], [8, 16, 3, 3, 是], [8, 17, 2, 2, 是], [8, 18, 1, 1, 是], [8, 19, 0, 0, 是], [8, 20, 26, 26, 是], [8, 21, 25, 25, 是], [8, 22, 24, 24, 是], [8, 23, 23, 23, 是], [8, 24, 22, 22, 是], [8, 25, 21, 21, 是], [8, 26, 20, 20, 是], [9, 0, 18, 18, 是], [9, 1, 17, 17, 是], [9, 2, 16, 16, 是], [9, 3, 15, 15, 是], [9, 4, 14, 14, 是], [9, 5, 13, 13, 是], [9, 6, 12, 12, 是], [9, 7, 11, 11, 是], [9, 8, 10, 10, 是], [9, 9, 9, 9, 是], [9, 10, 8, 8, 是], [9, 11, 7, 7, 是], [9, 12, 6, 6, 是], [9, 13, 5, 5, 是], [9, 14, 4, 4, 是], [9, 15, 3, 3, 是], [9, 16, 2, 2, 是], [9, 17, 1, 1, 是], [9, 18, 0, 0, 是], [9, 19, 26, 26, 是], [9, 20, 25, 25, 是], [9, 21, 24, 24, 是], [9, 22, 23, 23, 是], [9, 23, 22, 22, 是], [9, 24, 21, 21, 是], [9, 25, 20, 20, 是], [9, 26, 19, 19, 是], [10, 0, 17, 17, 是], [10, 1, 16, 16, 是], [10, 2, 15, 15, 是], [10, 3, 14, 14, 是], [10, 4, 13, 13, 是], [10, 5, 12, 12, 是], [10, 6, 11, 11, 是], [10, 7, 10, 10, 是], [10, 8, 9, 9, 是], [10, 9, 8, 8, 是], [10, 10, 7, 7, 是], [10, 11, 6, 6, 是], [10, 12, 5, 5, 是], [10, 13, 4, 4, 是], [10, 14, 3, 3, 是], [10, 15, 2, 2, 是], [10, 16, 1, 1, 是], [10, 17, 0, 0, 是], [10, 18, 26, 26, 是], [10, 19, 25, 25, 是], [10, 20, 24, 24, 是], [10, 21, 23, 23, 是], [10, 22, 22, 22, 是], [10, 23, 21, 21, 是], [10, 24, 20, 20, 是], [10, 25, 19, 19, 是], [10, 26, 18, 18, 是], [11, 0, 16, 16, 是], [11, 1, 15, 15, 是], [11, 2, 14, 14, 是], [11, 3, 13, 13, 是], [11, 4, 12, 12, 是], [11, 5, 11, 11, 是], [11, 6, 10, 10, 是], [11, 7, 9, 9, 是], [11, 8, 8, 8, 是], [11, 9, 7, 7, 是], [11, 10, 6, 6, 是], [11, 11, 5, 5, 是], [11, 12, 4, 4, 是], [11, 13, 3, 3, 是], [11, 14, 2, 2, 是], [11, 15, 1, 1, 是], [11, 16, 0, 0, 是], [11, 17, 26, 26, 是], [11, 18, 25, 25, 是], [11, 19, 24, 24, 是], [11, 20, 23, 23, 是], [11, 21, 22, 22, 是], [11, 22, 21, 21, 是], [11, 23, 20, 20, 是], [11, 24, 19, 19, 是], [11, 25, 18, 18, 是], [11, 26, 17, 17, 是], [12, 0, 15, 15, 是], [12, 1, 14, 14, 是], [12, 2, 13, 13, 是], [12, 3, 12, 12, 是], [12, 4, 11, 11, 是], [12, 5, 10, 10, 是], [12, 6, 9, 9, 是], [12, 7, 8, 8, 是], [12, 8, 7, 7, 是], [12, 9, 6, 6, 是], [12, 10, 5, 5, 是], [12, 11, 4, 4, 是], [12, 12, 3, 3, 是], [12, 13, 2, 2, 是], [12, 14, 1, 1, 是], [12, 15, 0, 0, 是], [12, 16, 26, 26, 是], [12, 17, 25, 25, 是], [12, 18, 24, 24, 是], [12, 19, 23, 23, 是], [12, 20, 22, 22, 是], [12, 21, 21, 21, 是], [12, 22, 20, 20, 是], [12, 23, 19, 19, 是], [12, 24, 18, 18, 是], [12, 25, 17, 17, 是], [12, 26, 16, 16, 是], [13, 0, 14, 14, 是], [13, 1, 13, 13, 是], [13, 2, 12, 12, 是], [13, 3, 11, 11, 是], [13, 4, 10, 10, 是], [13, 5, 9, 9, 是], [13, 6, 8, 8, 是], [13, 7, 7, 7, 是], [13, 8, 6, 6, 是], [13, 9, 5, 5, 是], [13, 10, 4, 4, 是], [13, 11, 3, 3, 是], [13, 12, 2, 2, 是], [13, 13, 1, 1, 是], [13, 14, 0, 0, 是], [13, 15, 26, 26, 是], [13, 16, 25, 25, 是], [13, 17, 24, 24, 是], [13, 18, 23, 23, 是], [13, 19, 22, 22, 是], [13, 20, 21, 21, 是], [13, 21, 20, 20, 是], [13, 22, 19, 19, 是], [13, 23, 18, 18, 是], [13, 24, 17, 17, 是], [13, 25, 16, 16, 是], [13, 26, 15, 15, 是], [14, 0, 13, 13, 是], [14, 1, 12, 12, 是], [14, 2, 11, 11, 是], [14, 3, 10, 10, 是], [14, 4, 9, 9, 是], [14, 5, 8, 8, 是], [14, 6, 7, 7, 是], [14, 7, 6, 6, 是], [14, 8, 5, 5, 是], [14, 9, 4, 4, 是], [14, 10, 3, 3, 是], [14, 11, 2, 2, 是], [14, 12, 1, 1, 是], [14, 13, 0, 0, 是], [14, 14, 26, 26, 是], [14, 15, 25, 25, 是], [14, 16, 24, 24, 是], [14, 17, 23, 23, 是], [14, 18, 22, 22, 是], [14, 19, 21, 21, 是], [14, 20, 20, 20, 是], [14, 21, 19, 19, 是], [14, 22, 18, 18, 是], [14, 23, 17, 17, 是], [14, 24, 16, 16, 是], [14, 25, 15, 15, 是], [14, 26, 14, 14, 是], [15, 0, 12, 12, 是], [15, 1, 11, 11, 是], [15, 2, 10, 10, 是], [15, 3, 9, 9, 是], [15, 4, 8, 8, 是], [15, 5, 7, 7, 是], [15, 6, 6, 6, 是], [15, 7, 5, 5, 是], [15, 8, 4, 4, 是], [15, 9, 3, 3, 是], [15, 10, 2, 2, 是], [15, 11, 1, 1, 是], [15, 12, 0, 0, 是], [15, 13, 26, 26, 是], [15, 14, 25, 25, 是], [15, 15, 24, 24, 是], [15, 16, 23, 23, 是], [15, 17, 22, 22, 是], [15, 18, 21, 21, 是], [15, 19, 20, 20, 是], [15, 20, 19, 19, 是], [15, 21, 18, 18, 是], [15, 22, 17, 17, 是], [15, 23, 16, 16, 是], [15, 24, 15, 15, 是], [15, 25, 14, 14, 是], [15, 26, 13, 13, 是], [16, 0, 11, 11, 是], [16, 1, 10, 10, 是], [16, 2, 9, 9, 是], [16, 3, 8, 8, 是], [16, 4, 7, 7, 是], [16, 5, 6, 6, 是], [16, 6, 5, 5, 是], [16, 7, 4, 4, 是], [16, 8, 3, 3, 是], [16, 9, 2, 2, 是], [16, 10, 1, 1, 是], [16, 11, 0, 0, 是], [16, 12, 26, 26, 是], [16, 13, 25, 25, 是], [16, 14, 24, 24, 是], [16, 15, 23, 23, 是], [16, 16, 22, 22, 是], [16, 17, 21, 21, 是], [16, 18, 20, 20, 是], [16, 19, 19, 19, 是], [16, 20, 18, 18, 是], [16, 21, 17, 17, 是], [16, 22, 16, 16, 是], [16, 23, 15, 15, 是], [16, 24, 14, 14, 是], [16, 25, 13, 13, 是], [16, 26, 12, 12, 是], [17, 0, 10, 10, 是], [17, 1, 9, 9, 是], [17, 2, 8, 8, 是], [17, 3, 7, 7, 是], [17, 4, 6, 6, 是], [17, 5, 5, 5, 是], [17, 6, 4, 4, 是], [17, 7, 3, 3, 是], [17, 8, 2, 2, 是], [17, 9, 1, 1, 是], [17, 10, 0, 0, 是], [17, 11, 26, 26, 是], [17, 12, 25, 25, 是], [17, 13, 24, 24, 是], [17, 14, 23, 23, 是], [17, 15, 22, 22, 是], [17, 16, 21, 21, 是], [17, 17, 20, 20, 是], [17, 18, 19, 19, 是], [17, 19, 18, 18, 是], [17, 20, 17, 17, 是], [17, 21, 16, 16, 是], [17, 22, 15, 15, 是], [17, 23, 14, 14, 是], [17, 24, 13, 13, 是], [17, 25, 12, 12, 是], [17, 26, 11, 11, 是], [18, 0, 9, 9, 是], [18, 1, 8, 8, 是], [18, 2, 7, 7, 是], [18, 3, 6, 6, 是], [18, 4, 5, 5, 是], [18, 5, 4, 4, 是], [18, 6, 3, 3, 是], [18, 7, 2, 2, 是], [18, 8, 1, 1, 是], [18, 9, 0, 0, 是], [18, 10, 26, 26, 是], [18, 11, 25, 25, 是], [18, 12, 24, 24, 是], [18, 13, 23, 23, 是], [18, 14, 22, 22, 是], [18, 15, 21, 21, 是], [18, 16, 20, 20, 是], [18, 17, 19, 19, 是], [18, 18, 18, 18, 是], [18, 19, 17, 17, 是], [18, 20, 16, 16, 是], [18, 21, 15, 15, 是], [18, 22, 14, 14, 是], [18, 23, 13, 13, 是], [18, 24, 12, 12, 是], [18, 25, 11, 11, 是], [18, 26, 10, 10, 是], [19, 0, 8, 8, 是], [19, 1, 7, 7, 是], [19, 2, 6, 6, 是], [19, 3, 5, 5, 是], [19, 4, 4, 4, 是], [19, 5, 3, 3, 是], [19, 6, 2, 2, 是], [19, 7, 1, 1, 是], [19, 8, 0, 0, 是], [19, 9, 26, 26, 是], [19, 10, 25, 25, 是], [19, 11, 24, 24, 是], [19, 12, 23, 23, 是], [19, 13, 22, 22, 是], [19, 14, 21, 21, 是], [19, 15, 20, 20, 是], [19, 16, 19, 19, 是], [19, 17, 18, 18, 是], [19, 18, 17, 17, 是], [19, 19, 16, 16, 是], [19, 20, 15, 15, 是], [19, 21, 14, 14, 是], [19, 22, 13, 13, 是], [19, 23, 12, 12, 是], [19, 24, 11, 11, 是], [19, 25, 10, 10, 是], [19, 26, 9, 9, 是], [20, 0, 7, 7, 是], [20, 1, 6, 6, 是], [20, 2, 5, 5, 是], [20, 3, 4, 4, 是], [20, 4, 3, 3, 是], [20, 5, 2, 2, 是], [20, 6, 1, 1, 是], [20, 7, 0, 0, 是], [20, 8, 26, 26, 是], [20, 9, 25, 25, 是], [20, 10, 24, 24, 是], [20, 11, 23, 23, 是], [20, 12, 22, 22, 是], [20, 13, 21, 21, 是], [20, 14, 20, 20, 是], [20, 15, 19, 19, 是], [20, 16, 18, 18, 是], [20, 17, 17, 17, 是], [20, 18, 16, 16, 是], [20, 19, 15, 15, 是], [20, 20, 14, 14, 是], [20, 21, 13, 13, 是], [20, 22, 12, 12, 是], [20, 23, 11, 11, 是], [20, 24, 10, 10, 是], [20, 25, 9, 9, 是], [20, 26, 8, 8, 是], [21, 0, 6, 6, 是], [21, 1, 5, 5, 是], [21, 2, 4, 4, 是], [21, 3, 3, 3, 是], [21, 4, 2, 2, 是], [21, 5, 1, 1, 是], [21, 6, 0, 0, 是], [21, 7, 26, 26, 是], [21, 8, 25, 25, 是], [21, 9, 24, 24, 是], [21, 10, 23, 23, 是], [21, 11, 22, 22, 是], [21, 12, 21, 21, 是], [21, 13, 20, 20, 是], [21, 14, 19, 19, 是], [21, 15, 18, 18, 是], [21, 16, 17, 17, 是], [21, 17, 16, 16, 是], [21, 18, 15, 15, 是], [21, 19, 14, 14, 是], [21, 20, 13, 13, 是], [21, 21, 12, 12, 是], [21, 22, 11, 11, 是], [21, 23, 10, 10, 是], [21, 24, 9, 9, 是], [21, 25, 8, 8, 是], [21, 26, 7, 7, 是], [22, 0, 5, 5, 是], [22, 1, 4, 4, 是], [22, 2, 3, 3, 是], [22, 3, 2, 2, 是], [22, 4, 1, 1, 是], [22, 5, 0, 0, 是], [22, 6, 26, 26, 是], [22, 7, 25, 25, 是], [22, 8, 24, 24, 是], [22, 9, 23, 23, 是], [22, 10, 22, 22, 是], [22, 11, 21, 21, 是], [22, 12, 20, 20, 是], [22, 13, 19, 19, 是], [22, 14, 18, 18, 是], [22, 15, 17, 17, 是], [22, 16, 16, 16, 是], [22, 17, 15, 15, 是], [22, 18, 14, 14, 是], [22, 19, 13, 13, 是], [22, 20, 12, 12, 是], [22, 21, 11, 11, 是], [22, 22, 10, 10, 是], [22, 23, 9, 9, 是], [22, 24, 8, 8, 是], [22, 25, 7, 7, 是], [22, 26, 6, 6, 是], [23, 0, 4, 4, 是], [23, 1, 3, 3, 是], [23, 2, 2, 2, 是], [23, 3, 1, 1, 是], [23, 4, 0, 0, 是], [23, 5, 26, 26, 是], [23, 6, 25, 25, 是], [23, 7, 24, 24, 是], [23, 8, 23, 23, 是], [23, 9, 22, 22, 是], [23, 10, 21, 21, 是], [23, 11, 20, 20, 是], [23, 12, 19, 19, 是], [23, 13, 18, 18, 是], [23, 14, 17, 17, 是], [23, 15, 16, 16, 是], [23, 16, 15, 15, 是], [23, 17, 14, 14, 是], [23, 18, 13, 13, 是], [23, 19, 12, 12, 是], [23, 20, 11, 11, 是], [23, 21, 10, 10, 是], [23, 22, 9, 9, 是], [23, 23, 8, 8, 是], [23, 24, 7, 7, 是], [23, 25, 6, 6, 是], [23, 26, 5, 5, 是], [24, 0, 3, 3, 是], [24, 1, 2, 2, 是], [24, 2, 1, 1, 是], [24, 3, 0, 0, 是], [24, 4, 26, 26, 是], [24, 5, 25, 25, 是], [24, 6, 24, 24, 是], [24, 7, 23, 23, 是], [24, 8, 22, 22, 是], [24, 9, 21, 21, 是], [24, 10, 20, 20, 是], [24, 11, 19, 19, 是], [24, 12, 18, 18, 是], [24, 13, 17, 17, 是], [24, 14, 16, 16, 是], [24, 15, 15, 15, 是], [24, 16, 14, 14, 是], [24, 17, 13, 13, 是], [24, 18, 12, 12, 是], [24, 19, 11, 11, 是], [24, 20, 10, 10, 是], [24, 21, 9, 9, 是], [24, 22, 8, 8, 是], [24, 23, 7, 7, 是], [24, 24, 6, 6, 是], [24, 25, 5, 5, 是], [24, 26, 4, 4, 是], [25, 0, 2, 2, 是], [25, 1, 1, 1, 是], [25, 2, 0, 0, 是], [25, 3, 26, 26, 是], [25, 4, 25, 25, 是], [25, 5, 24, 24, 是], [25, 6, 23, 23, 是], [25, 7, 22, 22, 是], [25, 8, 21, 21, 是], [25, 9, 20, 20, 是], [25, 10, 19, 19, 是], [25, 11, 18, 18, 是], [25, 12, 17, 17, 是], [25, 13, 16, 16, 是], [25, 14, 15, 15, 是], [25, 15, 14, 14, 是], [25, 16, 13, 13, 是], [25, 17, 12, 12, 是], [25, 18, 11, 11, 是], [25, 19, 10, 10, 是], [25, 20, 9, 9, 是], [25, 21, 8, 8, 是], [25, 22, 7, 7, 是], [25, 23, 6, 6, 是], [25, 24, 5, 5, 是], [25, 25, 4, 4, 是], [25, 26, 3, 3, 是], [26, 0, 1, 1, 是], [26, 1, 0, 0, 是], [26, 2, 26, 26, 是], [26, 3, 25, 25, 是], [26, 4, 24, 24, 是], [26, 5, 23, 23, 是], [26, 6, 22, 22, 是], [26, 7, 21, 21, 是], [26, 8, 20, 20, 是], [26, 9, 19, 19, 是], [26, 10, 18, 18, 是], [26, 11, 17, 17, 是], [26, 12, 16, 16, 是], [26, 13, 15, 15, 是], [26, 14, 14, 14, 是], [26, 15, 13, 13, 是], [26, 16, 12, 12, 是], [26, 17, 11, 11, 是], [26, 18, 10, 10, 是], [26, 19, 9, 9, 是], [26, 20, 8, 8, 是], [26, 21, 7, 7, 是], [26, 22, 6, 6, 是], [26, 23, 5, 5, 是], [26, 24, 4, 4, 是], [26, 25, 3, 3, 是], [26, 26, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)","explanation_zh":"Wave6第八層核對=(27, 27, [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, 是], [0, 1, 26, 26, 是], [0, 2, 25, 25, 是], [0, 3, 24, 24, 是], [0, 4, 23, 23, 是], [0, 5, 22, 22, 是], [0, 6, 21, 21, 是], [0, 7, 20, 20, 是], [0, 8, 19, 19, 是], [0, 9, 18, 18, 是], [0, 10, 17, 17, 是], [0, 11, 16, 16, 是], [0, 12, 15, 15, 是], [0, 13, 14, 14, 是], [0, 14, 13, 13, 是], [0, 15, 12, 12, 是], [0, 16, 11, 11, 是], [0, 17, 10, 10, 是], [0, 18, 9, 9, 是], [0, 19, 8, 8, 是], [0, 20, 7, 7, 是], [0, 21, 6, 6, 是], [0, 22, 5, 5, 是], [0, 23, 4, 4, 是], [0, 24, 3, 3, 是], [0, 25, 2, 2, 是], [0, 26, 1, 1, 是], [1, 0, 26, 26, 是], [1, 1, 25, 25, 是], [1, 2, 24, 24, 是], [1, 3, 23, 23, 是], [1, 4, 22, 22, 是], [1, 5, 21, 21, 是], [1, 6, 20, 20, 是], [1, 7, 19, 19, 是], [1, 8, 18, 18, 是], [1, 9, 17, 17, 是], [1, 10, 16, 16, 是], [1, 11, 15, 15, 是], [1, 12, 14, 14, 是], [1, 13, 13, 13, 是], [1, 14, 12, 12, 是], [1, 15, 11, 11, 是], [1, 16, 10, 10, 是], [1, 17, 9, 9, 是], [1, 18, 8, 8, 是], [1, 19, 7, 7, 是], [1, 20, 6, 6, 是], [1, 21, 5, 5, 是], [1, 22, 4, 4, 是], [1, 23, 3, 3, 是], [1, 24, 2, 2, 是], [1, 25, 1, 1, 是], [1, 26, 0, 0, 是], [2, 0, 25, 25, 是], [2, 1, 24, 24, 是], [2, 2, 23, 23, 是], [2, 3, 22, 22, 是], [2, 4, 21, 21, 是], [2, 5, 20, 20, 是], [2, 6, 19, 19, 是], [2, 7, 18, 18, 是], [2, 8, 17, 17, 是], [2, 9, 16, 16, 是], [2, 10, 15, 15, 是], [2, 11, 14, 14, 是], [2, 12, 13, 13, 是], [2, 13, 12, 12, 是], [2, 14, 11, 11, 是], [2, 15, 10, 10, 是], [2, 16, 9, 9, 是], [2, 17, 8, 8, 是], [2, 18, 7, 7, 是], [2, 19, 6, 6, 是], [2, 20, 5, 5, 是], [2, 21, 4, 4, 是], [2, 22, 3, 3, 是], [2, 23, 2, 2, 是], [2, 24, 1, 1, 是], [2, 25, 0, 0, 是], [2, 26, 26, 26, 是], [3, 0, 24, 24, 是], [3, 1, 23, 23, 是], [3, 2, 22, 22, 是], [3, 3, 21, 21, 是], [3, 4, 20, 20, 是], [3, 5, 19, 19, 是], [3, 6, 18, 18, 是], [3, 7, 17, 17, 是], [3, 8, 16, 16, 是], [3, 9, 15, 15, 是], [3, 10, 14, 14, 是], [3, 11, 13, 13, 是], [3, 12, 12, 12, 是], [3, 13, 11, 11, 是], [3, 14, 10, 10, 是], [3, 15, 9, 9, 是], [3, 16, 8, 8, 是], [3, 17, 7, 7, 是], [3, 18, 6, 6, 是], [3, 19, 5, 5, 是], [3, 20, 4, 4, 是], [3, 21, 3, 3, 是], [3, 22, 2, 2, 是], [3, 23, 1, 1, 是], [3, 24, 0, 0, 是], [3, 25, 26, 26, 是], [3, 26, 25, 25, 是], [4, 0, 23, 23, 是], [4, 1, 22, 22, 是], [4, 2, 21, 21, 是], [4, 3, 20, 20, 是], [4, 4, 19, 19, 是], [4, 5, 18, 18, 是], [4, 6, 17, 17, 是], [4, 7, 16, 16, 是], [4, 8, 15, 15, 是], [4, 9, 14, 14, 是], [4, 10, 13, 13, 是], [4, 11, 12, 12, 是], [4, 12, 11, 11, 是], [4, 13, 10, 10, 是], [4, 14, 9, 9, 是], [4, 15, 8, 8, 是], [4, 16, 7, 7, 是], [4, 17, 6, 6, 是], [4, 18, 5, 5, 是], [4, 19, 4, 4, 是], [4, 20, 3, 3, 是], [4, 21, 2, 2, 是], [4, 22, 1, 1, 是], [4, 23, 0, 0, 是], [4, 24, 26, 26, 是], [4, 25, 25, 25, 是], [4, 26, 24, 24, 是], [5, 0, 22, 22, 是], [5, 1, 21, 21, 是], [5, 2, 20, 20, 是], [5, 3, 19, 19, 是], [5, 4, 18, 18, 是], [5, 5, 17, 17, 是], [5, 6, 16, 16, 是], [5, 7, 15, 15, 是], [5, 8, 14, 14, 是], [5, 9, 13, 13, 是], [5, 10, 12, 12, 是], [5, 11, 11, 11, 是], [5, 12, 10, 10, 是], [5, 13, 9, 9, 是], [5, 14, 8, 8, 是], [5, 15, 7, 7, 是], [5, 16, 6, 6, 是], [5, 17, 5, 5, 是], [5, 18, 4, 4, 是], [5, 19, 3, 3, 是], [5, 20, 2, 2, 是], [5, 21, 1, 1, 是], [5, 22, 0, 0, 是], [5, 23, 26, 26, 是], [5, 24, 25, 25, 是], [5, 25, 24, 24, 是], [5, 26, 23, 23, 是], [6, 0, 21, 21, 是], [6, 1, 20, 20, 是], [6, 2, 19, 19, 是], [6, 3, 18, 18, 是], [6, 4, 17, 17, 是], [6, 5, 16, 16, 是], [6, 6, 15, 15, 是], [6, 7, 14, 14, 是], [6, 8, 13, 13, 是], [6, 9, 12, 12, 是], [6, 10, 11, 11, 是], [6, 11, 10, 10, 是], [6, 12, 9, 9, 是], [6, 13, 8, 8, 是], [6, 14, 7, 7, 是], [6, 15, 6, 6, 是], [6, 16, 5, 5, 是], [6, 17, 4, 4, 是], [6, 18, 3, 3, 是], [6, 19, 2, 2, 是], [6, 20, 1, 1, 是], [6, 21, 0, 0, 是], [6, 22, 26, 26, 是], [6, 23, 25, 25, 是], [6, 24, 24, 24, 是], [6, 25, 23, 23, 是], [6, 26, 22, 22, 是], [7, 0, 20, 20, 是], [7, 1, 19, 19, 是], [7, 2, 18, 18, 是], [7, 3, 17, 17, 是], [7, 4, 16, 16, 是], [7, 5, 15, 15, 是], [7, 6, 14, 14, 是], [7, 7, 13, 13, 是], [7, 8, 12, 12, 是], [7, 9, 11, 11, 是], [7, 10, 10, 10, 是], [7, 11, 9, 9, 是], [7, 12, 8, 8, 是], [7, 13, 7, 7, 是], [7, 14, 6, 6, 是], [7, 15, 5, 5, 是], [7, 16, 4, 4, 是], [7, 17, 3, 3, 是], [7, 18, 2, 2, 是], [7, 19, 1, 1, 是], [7, 20, 0, 0, 是], [7, 21, 26, 26, 是], [7, 22, 25, 25, 是], [7, 23, 24, 24, 是], [7, 24, 23, 23, 是], [7, 25, 22, 22, 是], [7, 26, 21, 21, 是], [8, 0, 19, 19, 是], [8, 1, 18, 18, 是], [8, 2, 17, 17, 是], [8, 3, 16, 16, 是], [8, 4, 15, 15, 是], [8, 5, 14, 14, 是], [8, 6, 13, 13, 是], [8, 7, 12, 12, 是], [8, 8, 11, 11, 是], [8, 9, 10, 10, 是], [8, 10, 9, 9, 是], [8, 11, 8, 8, 是], [8, 12, 7, 7, 是], [8, 13, 6, 6, 是], [8, 14, 5, 5, 是], [8, 15, 4, 4, 是], [8, 16, 3, 3, 是], [8, 17, 2, 2, 是], [8, 18, 1, 1, 是], [8, 19, 0, 0, 是], [8, 20, 26, 26, 是], [8, 21, 25, 25, 是], [8, 22, 24, 24, 是], [8, 23, 23, 23, 是], [8, 24, 22, 22, 是], [8, 25, 21, 21, 是], [8, 26, 20, 20, 是], [9, 0, 18, 18, 是], [9, 1, 17, 17, 是], [9, 2, 16, 16, 是], [9, 3, 15, 15, 是], [9, 4, 14, 14, 是], [9, 5, 13, 13, 是], [9, 6, 12, 12, 是], [9, 7, 11, 11, 是], [9, 8, 10, 10, 是], [9, 9, 9, 9, 是], [9, 10, 8, 8, 是], [9, 11, 7, 7, 是], [9, 12, 6, 6, 是], [9, 13, 5, 5, 是], [9, 14, 4, 4, 是], [9, 15, 3, 3, 是], [9, 16, 2, 2, 是], [9, 17, 1, 1, 是], [9, 18, 0, 0, 是], [9, 19, 26, 26, 是], [9, 20, 25, 25, 是], [9, 21, 24, 24, 是], [9, 22, 23, 23, 是], [9, 23, 22, 22, 是], [9, 24, 21, 21, 是], [9, 25, 20, 20, 是], [9, 26, 19, 19, 是], [10, 0, 17, 17, 是], [10, 1, 16, 16, 是], [10, 2, 15, 15, 是], [10, 3, 14, 14, 是], [10, 4, 13, 13, 是], [10, 5, 12, 12, 是], [10, 6, 11, 11, 是], [10, 7, 10, 10, 是], [10, 8, 9, 9, 是], [10, 9, 8, 8, 是], [10, 10, 7, 7, 是], [10, 11, 6, 6, 是], [10, 12, 5, 5, 是], [10, 13, 4, 4, 是], [10, 14, 3, 3, 是], [10, 15, 2, 2, 是], [10, 16, 1, 1, 是], [10, 17, 0, 0, 是], [10, 18, 26, 26, 是], [10, 19, 25, 25, 是], [10, 20, 24, 24, 是], [10, 21, 23, 23, 是], [10, 22, 22, 22, 是], [10, 23, 21, 21, 是], [10, 24, 20, 20, 是], [10, 25, 19, 19, 是], [10, 26, 18, 18, 是], [11, 0, 16, 16, 是], [11, 1, 15, 15, 是], [11, 2, 14, 14, 是], [11, 3, 13, 13, 是], [11, 4, 12, 12, 是], [11, 5, 11, 11, 是], [11, 6, 10, 10, 是], [11, 7, 9, 9, 是], [11, 8, 8, 8, 是], [11, 9, 7, 7, 是], [11, 10, 6, 6, 是], [11, 11, 5, 5, 是], [11, 12, 4, 4, 是], [11, 13, 3, 3, 是], [11, 14, 2, 2, 是], [11, 15, 1, 1, 是], [11, 16, 0, 0, 是], [11, 17, 26, 26, 是], [11, 18, 25, 25, 是], [11, 19, 24, 24, 是], [11, 20, 23, 23, 是], [11, 21, 22, 22, 是], [11, 22, 21, 21, 是], [11, 23, 20, 20, 是], [11, 24, 19, 19, 是], [11, 25, 18, 18, 是], [11, 26, 17, 17, 是], [12, 0, 15, 15, 是], [12, 1, 14, 14, 是], [12, 2, 13, 13, 是], [12, 3, 12, 12, 是], [12, 4, 11, 11, 是], [12, 5, 10, 10, 是], [12, 6, 9, 9, 是], [12, 7, 8, 8, 是], [12, 8, 7, 7, 是], [12, 9, 6, 6, 是], [12, 10, 5, 5, 是], [12, 11, 4, 4, 是], [12, 12, 3, 3, 是], [12, 13, 2, 2, 是], [12, 14, 1, 1, 是], [12, 15, 0, 0, 是], [12, 16, 26, 26, 是], [12, 17, 25, 25, 是], [12, 18, 24, 24, 是], [12, 19, 23, 23, 是], [12, 20, 22, 22, 是], [12, 21, 21, 21, 是], [12, 22, 20, 20, 是], [12, 23, 19, 19, 是], [12, 24, 18, 18, 是], [12, 25, 17, 17, 是], [12, 26, 16, 16, 是], [13, 0, 14, 14, 是], [13, 1, 13, 13, 是], [13, 2, 12, 12, 是], [13, 3, 11, 11, 是], [13, 4, 10, 10, 是], [13, 5, 9, 9, 是], [13, 6, 8, 8, 是], [13, 7, 7, 7, 是], [13, 8, 6, 6, 是], [13, 9, 5, 5, 是], [13, 10, 4, 4, 是], [13, 11, 3, 3, 是], [13, 12, 2, 2, 是], [13, 13, 1, 1, 是], [13, 14, 0, 0, 是], [13, 15, 26, 26, 是], [13, 16, 25, 25, 是], [13, 17, 24, 24, 是], [13, 18, 23, 23, 是], [13, 19, 22, 22, 是], [13, 20, 21, 21, 是], [13, 21, 20, 20, 是], [13, 22, 19, 19, 是], [13, 23, 18, 18, 是], [13, 24, 17, 17, 是], [13, 25, 16, 16, 是], [13, 26, 15, 15, 是], [14, 0, 13, 13, 是], [14, 1, 12, 12, 是], [14, 2, 11, 11, 是], [14, 3, 10, 10, 是], [14, 4, 9, 9, 是], [14, 5, 8, 8, 是], [14, 6, 7, 7, 是], [14, 7, 6, 6, 是], [14, 8, 5, 5, 是], [14, 9, 4, 4, 是], [14, 10, 3, 3, 是], [14, 11, 2, 2, 是], [14, 12, 1, 1, 是], [14, 13, 0, 0, 是], [14, 14, 26, 26, 是], [14, 15, 25, 25, 是], [14, 16, 24, 24, 是], [14, 17, 23, 23, 是], [14, 18, 22, 22, 是], [14, 19, 21, 21, 是], [14, 20, 20, 20, 是], [14, 21, 19, 19, 是], [14, 22, 18, 18, 是], [14, 23, 17, 17, 是], [14, 24, 16, 16, 是], [14, 25, 15, 15, 是], [14, 26, 14, 14, 是], [15, 0, 12, 12, 是], [15, 1, 11, 11, 是], [15, 2, 10, 10, 是], [15, 3, 9, 9, 是], [15, 4, 8, 8, 是], [15, 5, 7, 7, 是], [15, 6, 6, 6, 是], [15, 7, 5, 5, 是], [15, 8, 4, 4, 是], [15, 9, 3, 3, 是], [15, 10, 2, 2, 是], [15, 11, 1, 1, 是], [15, 12, 0, 0, 是], [15, 13, 26, 26, 是], [15, 14, 25, 25, 是], [15, 15, 24, 24, 是], [15, 16, 23, 23, 是], [15, 17, 22, 22, 是], [15, 18, 21, 21, 是], [15, 19, 20, 20, 是], [15, 20, 19, 19, 是], [15, 21, 18, 18, 是], [15, 22, 17, 17, 是], [15, 23, 16, 16, 是], [15, 24, 15, 15, 是], [15, 25, 14, 14, 是], [15, 26, 13, 13, 是], [16, 0, 11, 11, 是], [16, 1, 10, 10, 是], [16, 2, 9, 9, 是], [16, 3, 8, 8, 是], [16, 4, 7, 7, 是], [16, 5, 6, 6, 是], [16, 6, 5, 5, 是], [16, 7, 4, 4, 是], [16, 8, 3, 3, 是], [16, 9, 2, 2, 是], [16, 10, 1, 1, 是], [16, 11, 0, 0, 是], [16, 12, 26, 26, 是], [16, 13, 25, 25, 是], [16, 14, 24, 24, 是], [16, 15, 23, 23, 是], [16, 16, 22, 22, 是], [16, 17, 21, 21, 是], [16, 18, 20, 20, 是], [16, 19, 19, 19, 是], [16, 20, 18, 18, 是], [16, 21, 17, 17, 是], [16, 22, 16, 16, 是], [16, 23, 15, 15, 是], [16, 24, 14, 14, 是], [16, 25, 13, 13, 是], [16, 26, 12, 12, 是], [17, 0, 10, 10, 是], [17, 1, 9, 9, 是], [17, 2, 8, 8, 是], [17, 3, 7, 7, 是], [17, 4, 6, 6, 是], [17, 5, 5, 5, 是], [17, 6, 4, 4, 是], [17, 7, 3, 3, 是], [17, 8, 2, 2, 是], [17, 9, 1, 1, 是], [17, 10, 0, 0, 是], [17, 11, 26, 26, 是], [17, 12, 25, 25, 是], [17, 13, 24, 24, 是], [17, 14, 23, 23, 是], [17, 15, 22, 22, 是], [17, 16, 21, 21, 是], [17, 17, 20, 20, 是], [17, 18, 19, 19, 是], [17, 19, 18, 18, 是], [17, 20, 17, 17, 是], [17, 21, 16, 16, 是], [17, 22, 15, 15, 是], [17, 23, 14, 14, 是], [17, 24, 13, 13, 是], [17, 25, 12, 12, 是], [17, 26, 11, 11, 是], [18, 0, 9, 9, 是], [18, 1, 8, 8, 是], [18, 2, 7, 7, 是], [18, 3, 6, 6, 是], [18, 4, 5, 5, 是], [18, 5, 4, 4, 是], [18, 6, 3, 3, 是], [18, 7, 2, 2, 是], [18, 8, 1, 1, 是], [18, 9, 0, 0, 是], [18, 10, 26, 26, 是], [18, 11, 25, 25, 是], [18, 12, 24, 24, 是], [18, 13, 23, 23, 是], [18, 14, 22, 22, 是], [18, 15, 21, 21, 是], [18, 16, 20, 20, 是], [18, 17, 19, 19, 是], [18, 18, 18, 18, 是], [18, 19, 17, 17, 是], [18, 20, 16, 16, 是], [18, 21, 15, 15, 是], [18, 22, 14, 14, 是], [18, 23, 13, 13, 是], [18, 24, 12, 12, 是], [18, 25, 11, 11, 是], [18, 26, 10, 10, 是], [19, 0, 8, 8, 是], [19, 1, 7, 7, 是], [19, 2, 6, 6, 是], [19, 3, 5, 5, 是], [19, 4, 4, 4, 是], [19, 5, 3, 3, 是], [19, 6, 2, 2, 是], [19, 7, 1, 1, 是], [19, 8, 0, 0, 是], [19, 9, 26, 26, 是], [19, 10, 25, 25, 是], [19, 11, 24, 24, 是], [19, 12, 23, 23, 是], [19, 13, 22, 22, 是], [19, 14, 21, 21, 是], [19, 15, 20, 20, 是], [19, 16, 19, 19, 是], [19, 17, 18, 18, 是], [19, 18, 17, 17, 是], [19, 19, 16, 16, 是], [19, 20, 15, 15, 是], [19, 21, 14, 14, 是], [19, 22, 13, 13, 是], [19, 23, 12, 12, 是], [19, 24, 11, 11, 是], [19, 25, 10, 10, 是], [19, 26, 9, 9, 是], [20, 0, 7, 7, 是], [20, 1, 6, 6, 是], [20, 2, 5, 5, 是], [20, 3, 4, 4, 是], [20, 4, 3, 3, 是], [20, 5, 2, 2, 是], [20, 6, 1, 1, 是], [20, 7, 0, 0, 是], [20, 8, 26, 26, 是], [20, 9, 25, 25, 是], [20, 10, 24, 24, 是], [20, 11, 23, 23, 是], [20, 12, 22, 22, 是], [20, 13, 21, 21, 是], [20, 14, 20, 20, 是], [20, 15, 19, 19, 是], [20, 16, 18, 18, 是], [20, 17, 17, 17, 是], [20, 18, 16, 16, 是], [20, 19, 15, 15, 是], [20, 20, 14, 14, 是], [20, 21, 13, 13, 是], [20, 22, 12, 12, 是], [20, 23, 11, 11, 是], [20, 24, 10, 10, 是], [20, 25, 9, 9, 是], [20, 26, 8, 8, 是], [21, 0, 6, 6, 是], [21, 1, 5, 5, 是], [21, 2, 4, 4, 是], [21, 3, 3, 3, 是], [21, 4, 2, 2, 是], [21, 5, 1, 1, 是], [21, 6, 0, 0, 是], [21, 7, 26, 26, 是], [21, 8, 25, 25, 是], [21, 9, 24, 24, 是], [21, 10, 23, 23, 是], [21, 11, 22, 22, 是], [21, 12, 21, 21, 是], [21, 13, 20, 20, 是], [21, 14, 19, 19, 是], [21, 15, 18, 18, 是], [21, 16, 17, 17, 是], [21, 17, 16, 16, 是], [21, 18, 15, 15, 是], [21, 19, 14, 14, 是], [21, 20, 13, 13, 是], [21, 21, 12, 12, 是], [21, 22, 11, 11, 是], [21, 23, 10, 10, 是], [21, 24, 9, 9, 是], [21, 25, 8, 8, 是], [21, 26, 7, 7, 是], [22, 0, 5, 5, 是], [22, 1, 4, 4, 是], [22, 2, 3, 3, 是], [22, 3, 2, 2, 是], [22, 4, 1, 1, 是], [22, 5, 0, 0, 是], [22, 6, 26, 26, 是], [22, 7, 25, 25, 是], [22, 8, 24, 24, 是], [22, 9, 23, 23, 是], [22, 10, 22, 22, 是], [22, 11, 21, 21, 是], [22, 12, 20, 20, 是], [22, 13, 19, 19, 是], [22, 14, 18, 18, 是], [22, 15, 17, 17, 是], [22, 16, 16, 16, 是], [22, 17, 15, 15, 是], [22, 18, 14, 14, 是], [22, 19, 13, 13, 是], [22, 20, 12, 12, 是], [22, 21, 11, 11, 是], [22, 22, 10, 10, 是], [22, 23, 9, 9, 是], [22, 24, 8, 8, 是], [22, 25, 7, 7, 是], [22, 26, 6, 6, 是], [23, 0, 4, 4, 是], [23, 1, 3, 3, 是], [23, 2, 2, 2, 是], [23, 3, 1, 1, 是], [23, 4, 0, 0, 是], [23, 5, 26, 26, 是], [23, 6, 25, 25, 是], [23, 7, 24, 24, 是], [23, 8, 23, 23, 是], [23, 9, 22, 22, 是], [23, 10, 21, 21, 是], [23, 11, 20, 20, 是], [23, 12, 19, 19, 是], [23, 13, 18, 18, 是], [23, 14, 17, 17, 是], [23, 15, 16, 16, 是], [23, 16, 15, 15, 是], [23, 17, 14, 14, 是], [23, 18, 13, 13, 是], [23, 19, 12, 12, 是], [23, 20, 11, 11, 是], [23, 21, 10, 10, 是], [23, 22, 9, 9, 是], [23, 23, 8, 8, 是], [23, 24, 7, 7, 是], [23, 25, 6, 6, 是], [23, 26, 5, 5, 是], [24, 0, 3, 3, 是], [24, 1, 2, 2, 是], [24, 2, 1, 1, 是], [24, 3, 0, 0, 是], [24, 4, 26, 26, 是], [24, 5, 25, 25, 是], [24, 6, 24, 24, 是], [24, 7, 23, 23, 是], [24, 8, 22, 22, 是], [24, 9, 21, 21, 是], [24, 10, 20, 20, 是], [24, 11, 19, 19, 是], [24, 12, 18, 18, 是], [24, 13, 17, 17, 是], [24, 14, 16, 16, 是], [24, 15, 15, 15, 是], [24, 16, 14, 14, 是], [24, 17, 13, 13, 是], [24, 18, 12, 12, 是], [24, 19, 11, 11, 是], [24, 20, 10, 10, 是], [24, 21, 9, 9, 是], [24, 22, 8, 8, 是], [24, 23, 7, 7, 是], [24, 24, 6, 6, 是], [24, 25, 5, 5, 是], [24, 26, 4, 4, 是], [25, 0, 2, 2, 是], [25, 1, 1, 1, 是], [25, 2, 0, 0, 是], [25, 3, 26, 26, 是], [25, 4, 25, 25, 是], [25, 5, 24, 24, 是], [25, 6, 23, 23, 是], [25, 7, 22, 22, 是], [25, 8, 21, 21, 是], [25, 9, 20, 20, 是], [25, 10, 19, 19, 是], [25, 11, 18, 18, 是], [25, 12, 17, 17, 是], [25, 13, 16, 16, 是], [25, 14, 15, 15, 是], [25, 15, 14, 14, 是], [25, 16, 13, 13, 是], [25, 17, 12, 12, 是], [25, 18, 11, 11, 是], [25, 19, 10, 10, 是], [25, 20, 9, 9, 是], [25, 21, 8, 8, 是], [25, 22, 7, 7, 是], [25, 23, 6, 6, 是], [25, 24, 5, 5, 是], [25, 25, 4, 4, 是], [25, 26, 3, 3, 是], [26, 0, 1, 1, 是], [26, 1, 0, 0, 是], [26, 2, 26, 26, 是], [26, 3, 25, 25, 是], [26, 4, 24, 24, 是], [26, 5, 23, 23, 是], [26, 6, 22, 22, 是], [26, 7, 21, 21, 是], [26, 8, 20, 20, 是], [26, 9, 19, 19, 是], [26, 10, 18, 18, 是], [26, 11, 17, 17, 是], [26, 12, 16, 16, 是], [26, 13, 15, 15, 是], [26, 14, 14, 14, 是], [26, 15, 13, 13, 是], [26, 16, 12, 12, 是], [26, 17, 11, 11, 是], [26, 18, 10, 10, 是], [26, 19, 9, 9, 是], [26, 20, 8, 8, 是], [26, 21, 7, 7, 是], [26, 22, 6, 6, 是], [26, 23, 5, 5, 是], [26, 24, 4, 4, 是], [26, 25, 3, 3, 是], [26, 26, 2, 2, 是]], 是, null, complete_total_map_satisfies_every_product_pair)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141025,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],"target_identity":0,"mapping":[0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-homomorphism-qualification","semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0026","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], 是, 是, 是, [[0, [0, 2], 是], [1, [0, 2], 是], [2, [0, 2], 是], [3, [0, 2], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2], 是, 是, 是, [[0, [0, 2], 是], [1, [0, 2], 是], [2, [0, 2], 是], [3, [0, 2], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141026,"parameters":{"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0027","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4], 是, 是, 是, [[0, [0, 2, 4], 是], [1, [0, 2, 4], 是], [2, [0, 2, 4], 是], [3, [0, 2, 4], 是], [4, [0, 2, 4], 是], [5, [0, 2, 4], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4], 是, 是, 是, [[0, [0, 2, 4], 是], [1, [0, 2, 4], 是], [2, [0, 2, 4], 是], [3, [0, 2, 4], 是], [4, [0, 2, 4], 是], [5, [0, 2, 4], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141027,"parameters":{"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0028","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6], 是, 是, 是, [[0, [0, 2, 4, 6], 是], [1, [0, 2, 4, 6], 是], [2, [0, 2, 4, 6], 是], [3, [0, 2, 4, 6], 是], [4, [0, 2, 4, 6], 是], [5, [0, 2, 4, 6], 是], [6, [0, 2, 4, 6], 是], [7, [0, 2, 4, 6], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6], 是, 是, 是, [[0, [0, 2, 4, 6], 是], [1, [0, 2, 4, 6], 是], [2, [0, 2, 4, 6], 是], [3, [0, 2, 4, 6], 是], [4, [0, 2, 4, 6], 是], [5, [0, 2, 4, 6], 是], [6, [0, 2, 4, 6], 是], [7, [0, 2, 4, 6], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141028,"parameters":{"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0029","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8], 是, 是, 是, [[0, [0, 2, 4, 6, 8], 是], [1, [0, 2, 4, 6, 8], 是], [2, [0, 2, 4, 6, 8], 是], [3, [0, 2, 4, 6, 8], 是], [4, [0, 2, 4, 6, 8], 是], [5, [0, 2, 4, 6, 8], 是], [6, [0, 2, 4, 6, 8], 是], [7, [0, 2, 4, 6, 8], 是], [8, [0, 2, 4, 6, 8], 是], [9, [0, 2, 4, 6, 8], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8], 是, 是, 是, [[0, [0, 2, 4, 6, 8], 是], [1, [0, 2, 4, 6, 8], 是], [2, [0, 2, 4, 6, 8], 是], [3, [0, 2, 4, 6, 8], 是], [4, [0, 2, 4, 6, 8], 是], [5, [0, 2, 4, 6, 8], 是], [6, [0, 2, 4, 6, 8], 是], [7, [0, 2, 4, 6, 8], 是], [8, [0, 2, 4, 6, 8], 是], [9, [0, 2, 4, 6, 8], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141029,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0030","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10], 是], [1, [0, 2, 4, 6, 8, 10], 是], [2, [0, 2, 4, 6, 8, 10], 是], [3, [0, 2, 4, 6, 8, 10], 是], [4, [0, 2, 4, 6, 8, 10], 是], [5, [0, 2, 4, 6, 8, 10], 是], [6, [0, 2, 4, 6, 8, 10], 是], [7, [0, 2, 4, 6, 8, 10], 是], [8, [0, 2, 4, 6, 8, 10], 是], [9, [0, 2, 4, 6, 8, 10], 是], [10, [0, 2, 4, 6, 8, 10], 是], [11, [0, 2, 4, 6, 8, 10], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10], 是], [1, [0, 2, 4, 6, 8, 10], 是], [2, [0, 2, 4, 6, 8, 10], 是], [3, [0, 2, 4, 6, 8, 10], 是], [4, [0, 2, 4, 6, 8, 10], 是], [5, [0, 2, 4, 6, 8, 10], 是], [6, [0, 2, 4, 6, 8, 10], 是], [7, [0, 2, 4, 6, 8, 10], 是], [8, [0, 2, 4, 6, 8, 10], 是], [9, [0, 2, 4, 6, 8, 10], 是], [10, [0, 2, 4, 6, 8, 10], 是], [11, [0, 2, 4, 6, 8, 10], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141030,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0031","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12], 是], [1, [0, 2, 4, 6, 8, 10, 12], 是], [2, [0, 2, 4, 6, 8, 10, 12], 是], [3, [0, 2, 4, 6, 8, 10, 12], 是], [4, [0, 2, 4, 6, 8, 10, 12], 是], [5, [0, 2, 4, 6, 8, 10, 12], 是], [6, [0, 2, 4, 6, 8, 10, 12], 是], [7, [0, 2, 4, 6, 8, 10, 12], 是], [8, [0, 2, 4, 6, 8, 10, 12], 是], [9, [0, 2, 4, 6, 8, 10, 12], 是], [10, [0, 2, 4, 6, 8, 10, 12], 是], [11, [0, 2, 4, 6, 8, 10, 12], 是], [12, [0, 2, 4, 6, 8, 10, 12], 是], [13, [0, 2, 4, 6, 8, 10, 12], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12], 是], [1, [0, 2, 4, 6, 8, 10, 12], 是], [2, [0, 2, 4, 6, 8, 10, 12], 是], [3, [0, 2, 4, 6, 8, 10, 12], 是], [4, [0, 2, 4, 6, 8, 10, 12], 是], [5, [0, 2, 4, 6, 8, 10, 12], 是], [6, [0, 2, 4, 6, 8, 10, 12], 是], [7, [0, 2, 4, 6, 8, 10, 12], 是], [8, [0, 2, 4, 6, 8, 10, 12], 是], [9, [0, 2, 4, 6, 8, 10, 12], 是], [10, [0, 2, 4, 6, 8, 10, 12], 是], [11, [0, 2, 4, 6, 8, 10, 12], 是], [12, [0, 2, 4, 6, 8, 10, 12], 是], [13, [0, 2, 4, 6, 8, 10, 12], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141031,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0032","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141032,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0033","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141033,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0034","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141034,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0035","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141035,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0036","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141036,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0037","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141037,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0038","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141038,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0039","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141039,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0040","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141040,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0041","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141041,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0042","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141042,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0043","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141043,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0044","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141044,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0045","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141045,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0046","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141046,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16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Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0047","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141047,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,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Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0048","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], 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42, 44, 46], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141048,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-normal-subgroup-audit","semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17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Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0049","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141049,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,4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le\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0050","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-normal-subgroup-audit；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,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exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [50, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [51, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation_zh":"Wave6第八層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是, 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [50, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是], [51, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是]], 是, 是, identity_preimage_is_a_normal_subgroup_by_complete_audit)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141050,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,2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50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0051","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1],\"a\":0,\"b\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 0, 0, 是, 2, 是, [0, 2], [0, 2], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(0, 2, 0, 0, 是, 2, 是, [0, 2], [0, 2], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141051,"parameters":{"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1],"a":0,"b":2,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":0,\"b\":2,\"mapping\":[0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0052","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1],\"a\":1,\"b\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 1, 1, 是, 2, 是, [1, 3, 5], [1, 3, 5], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(1, 3, 1, 1, 是, 2, 是, [1, 3, 5], [1, 3, 5], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141052,"parameters":{"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1],"a":1,"b":3,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":1,\"b\":3,\"mapping\":[0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0053","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1],\"a\":2,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 0, 0, 是, 2, 是, [0, 2, 4, 6], [0, 2, 4, 6], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(2, 4, 0, 0, 是, 2, 是, [0, 2, 4, 6], [0, 2, 4, 6], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141053,"parameters":{"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1],"a":2,"b":4,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":2,\"b\":4,\"mapping\":[0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0054","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1],\"a\":3,\"b\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9], [1, 3, 5, 7, 9], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(3, 5, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9], [1, 3, 5, 7, 9], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141054,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1],"a":3,"b":5,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":3,\"b\":5,\"mapping\":[0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0055","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1],\"a\":4,\"b\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 6, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(4, 6, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141055,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1],"a":4,"b":6,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":4,\"b\":6,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0056","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":5,\"b\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 7, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13], [1, 3, 5, 7, 9, 11, 13], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(5, 7, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13], [1, 3, 5, 7, 9, 11, 13], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141056,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":5,"b":7,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":5,\"b\":7,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0057","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":6,\"b\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 8, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(6, 8, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141057,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":6,"b":8,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":6,\"b\":8,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0058","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":7,\"b\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 9, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17], [1, 3, 5, 7, 9, 11, 13, 15, 17], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(7, 9, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17], [1, 3, 5, 7, 9, 11, 13, 15, 17], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141058,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":7,"b":9,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":7,\"b\":9,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0059","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":8,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 10, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(8, 10, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141059,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":8,"b":10,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":8,\"b\":10,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0060","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":9,\"b\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 11, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(9, 11, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141060,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":9,"b":11,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":9,\"b\":11,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0061","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":10,\"b\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 12, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(10, 12, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141061,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":10,"b":12,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":10,\"b\":12,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0062","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":11,\"b\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 13, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(11, 13, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141062,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":11,"b":13,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":11,\"b\":13,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0063","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":12,\"b\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 14, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(12, 14, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141063,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":12,"b":14,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":12,\"b\":14,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0064","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":13,\"b\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 15, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(13, 15, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141064,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":13,"b":15,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":13,\"b\":15,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0065","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":14,\"b\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 16, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(14, 16, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141065,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":14,"b":16,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":14,\"b\":16,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0066","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":15,\"b\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 17, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(15, 17, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141066,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":15,"b":17,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":15,\"b\":17,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0067","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":16,\"b\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 18, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(16, 18, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141067,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":16,"b":18,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":16,\"b\":18,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0068","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":17,\"b\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 19, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(17, 19, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141068,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":17,"b":19,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":17,\"b\":19,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0069","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":18,\"b\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 20, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(18, 20, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141069,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":18,"b":20,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":18,\"b\":20,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0070","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":19,\"b\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 21, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(19, 21, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141070,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"a":19,"b":21,"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-fiber-coset-equivalence","semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":19,\"b\":21,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0071","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":20,\"b\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 22, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(20, 22, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141071,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,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Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0072","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":21,\"b\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 23, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(21, 23, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141072,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,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Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0073","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":22,\"b\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 24, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(22, 24, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141073,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22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Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0074","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":23,\"b\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 25, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(23, 25, 1, 1, 是, 2, 是, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141074,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45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8]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0075","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-fiber-coset-equivalence；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]],\"source_identity\":0,\"target_table\":[[0,1],[1,0]],\"target_identity\":0,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"a\":24,\"b\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 26, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation_zh":"Wave6第八層核對=(24, 26, 0, 0, 是, 2, 是, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 是, 是, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141075,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27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Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0076","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2],[1,2,0],[2,0,1]],\"source_identity\":0,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]],\"target_identity\":0,\"mapping\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141076,"parameters":{"source_table":[[0,1,2],[1,2,0],[2,0,1]],"source_identity":0,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"target_identity":0,"mapping":[0,1,2],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2],\"source_identity\":0,\"source_table\":[[0,1,2],[1,2,0],[2,0,1]],\"target_identity\":0,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0077","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"source_identity\":0,\"target_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"mapping\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141077,"parameters":{"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,1,2,3],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3],\"source_identity\":0,\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"target_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0078","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_identity\":0,\"mapping\":[0,1,2,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141078,"parameters":{"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,1,2,3,4],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4],\"source_identity\":0,\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0079","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141079,"parameters":{"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,1,2,3,4,5],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0080","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141080,"parameters":{"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,1,2,3,4,5,6],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0081","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141081,"parameters":{"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0082","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141082,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0083","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141083,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0084","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141084,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0085","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141085,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0086","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141086,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0087","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141087,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0088","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141088,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0089","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141089,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0090","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141090,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0091","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141091,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0092","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141092,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0093","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141093,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0094","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141094,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0095","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141095,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0096","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141096,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0097","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141097,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0098","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141098,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0099","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141099,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0100","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-homomorphism-kernel"],"title_zh":"有限群同態的核","stem_zh":"完整來源／目標Cayley table與complete total map逐對同態核對。操作=kernel-trivial-injective-boundary；資料={\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],\"target_identity\":0,\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation_zh":"Wave6第八層核對=([0], 是, 是, 是, 否, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)；依完整明示資料獨立重算；kernel是目標identity的完整preimage，subgroup、forall g,k的normality、fibers與injectivity均完整枚舉；非同態不得套kernel-normality；本節點不宣稱quotient或第一同構定理。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群同態的核"]},"variant":{"family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141100,"parameters":{"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],"domain":"Complete_finite_group_homomorphism_kernel","operation":"kernel-trivial-injective-boundary","semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}