{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r02-finite-group-center-0001","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 否, 否, 否]], [2, [是, 是, 是, 否, 否, 否]], [3, [是, 否, 否, 是, 否, 否]], [4, [是, 否, 否, 否, 是, 否]], [5, [是, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 否, 否, 否]], [2, [是, 是, 是, 否, 否, 否]], [3, [是, 否, 否, 是, 否, 否]], [4, [是, 否, 否, 否, 是, 否]], [5, [是, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152001,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0002","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是]], [3, [是, 是, 是, 是, 否, 否, 否, 否]], [4, [是, 否, 是, 否, 是, 否, 是, 否]], [5, [是, 否, 是, 否, 否, 是, 否, 是]], [6, [是, 否, 是, 否, 是, 否, 是, 否]], [7, [是, 否, 是, 否, 否, 是, 否, 是]]], [0, 2], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是]], [3, [是, 是, 是, 是, 否, 否, 否, 否]], [4, [是, 否, 是, 否, 是, 否, 是, 否]], [5, [是, 否, 是, 否, 否, 是, 否, 是]], [6, [是, 否, 是, 否, 是, 否, 是, 否]], [7, [是, 否, 是, 否, 否, 是, 否, 是]]], [0, 2], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152002,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0003","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 否, 否, 否, 否, 否]], [5, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [6, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [7, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [8, [是, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [9, [是, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 否, 否, 否, 否, 否]], [5, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [6, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [7, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [8, [是, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [9, [是, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152003,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0004","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [4, [是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否]], [6, [是, 否, 否, 是, 否, 否, 是, 否, 否, 是, 否, 否]], [7, [是, 否, 否, 是, 否, 否, 否, 是, 否, 否, 是, 否]], [8, [是, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 是]], [9, [是, 否, 否, 是, 否, 否, 是, 否, 否, 是, 否, 否]], [10, [是, 否, 否, 是, 否, 否, 否, 是, 否, 否, 是, 否]], [11, [是, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 是]]], [0, 3], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [4, [是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否]], [6, [是, 否, 否, 是, 否, 否, 是, 否, 否, 是, 否, 否]], [7, [是, 否, 否, 是, 否, 否, 否, 是, 否, 否, 是, 否]], [8, [是, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 是]], [9, [是, 否, 否, 是, 否, 否, 是, 否, 否, 是, 否, 否]], [10, [是, 否, 否, 是, 否, 否, 否, 是, 否, 否, 是, 否]], [11, [是, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 是]]], [0, 3], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152004,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0005","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [8, [是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [9, [是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [10, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [11, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [12, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [13, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [8, [是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [9, [是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [10, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [11, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [12, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [13, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152005,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0006","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否, 否]], [9, [是, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否]], [10, [是, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 是, 否]], [11, [是, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 是]], [12, [是, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否, 否]], [13, [是, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否]], [14, [是, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 是, 否]], [15, [是, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 是]]], [0, 4], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否, 否]], [9, [是, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否]], [10, [是, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 是, 否]], [11, [是, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 是]], [12, [是, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否, 否]], [13, [是, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 是, 否, 否]], [14, [是, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 是, 否]], [15, [是, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 是]]], [0, 4], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152006,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0007","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [12, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [13, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [17, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [12, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [13, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [17, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152007,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0008","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [11, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否]], [12, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否]], [13, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否]], [14, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是]], [15, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [16, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否]], [17, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否]], [18, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否]], [19, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是]]], [0, 5], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [11, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否]], [12, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否]], [13, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否]], [14, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是]], [15, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [16, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否, 否]], [17, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否, 否]], [18, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是, 否]], [19, [是, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 是]]], [0, 5], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152008,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0009","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [12, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [13, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [17, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [18, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [19, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [20, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [21, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [12, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [13, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [17, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [18, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [19, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [20, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [21, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152009,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0010","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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是]]], [0, 6], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [12, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [13, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否]], [17, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是]], [18, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [19, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [20, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [21, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否, 否]], [22, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是, 否]], [23, [是, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 是]]], [0, 6], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152010,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0011","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [12, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [13, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [17, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [18, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否]], [19, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [20, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [21, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [22, [是, 否, 否, 否, 否, 否, 否, 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否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [12, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [13, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [17, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否]], [18, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否]], [19, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [20, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [21, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [22, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [23, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [24, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [25, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152011,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0012","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [12, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [13, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [17, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [18, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [19, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否]], [20, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是]], [21, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [22, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [23, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [24, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [25, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [26, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否]], [27, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是]]], [0, 7], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [12, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [13, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [14, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [15, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [16, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [17, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [18, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [19, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否]], [20, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是]], [21, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否]], [22, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否]], [23, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否]], [24, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [25, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [26, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是, 否]], [27, [是, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 是]]], [0, 7], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152012,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0013","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [9, [是, 是, 是, 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否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152013,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0014","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [9, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [10, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [11, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [12, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [13, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [14, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152014,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0015","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [8, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152015,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0016","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 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否, 否, 否, 否, 否, 是, 否, 否]], [34, [是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [35, [是, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0, 9], Z_G_contains_exactly_the_elements_commuting_with_every_g)","explanation_zh":"Wave6第九層核對=([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 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equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152016,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0017","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否]], [35, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否]], [36, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [37, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152017,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0018","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [7, [是, 是, 是, 是, 是, 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equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152018,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-commutation-table","semantic_template":"qf-mod46-center-commutation-table/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0019","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 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equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152019,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0020","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12],[34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 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equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152020,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12],[34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,16,15,14,13,12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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0021","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10],[33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11],[34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12],[35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152021,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10],[33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11],[34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12],[35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0022","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11],[35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12],[36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [6, 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equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152022,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11],[35,34,33,32,31,30,29,28,27,26,25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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0023","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,9,8,7,6,5,4,3,2,1,0,24,23,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Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152023,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,8,7,6,5,4,3,2,1,0,24,23,22,21,2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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0024","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 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否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否]], [51, [是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 是]]], [0, 13], Z_G_contains_exactly_the_elements_commuting_with_every_g)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152024,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0025","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-commutation-table；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[3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exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], [1, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [2, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [3, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [4, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否, 否]], [5, [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 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equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-commutation-table","seed":202611152025,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,28,27,53,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4],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25],[52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26],[53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0026","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152026,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0027","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 2], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152027,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0028","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152028,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0029","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 3], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 3], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152029,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0030","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152030,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0031","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 4], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 4], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152031,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0032","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152032,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0033","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 5], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 5], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152033,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0034","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152034,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0035","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 6], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 6], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152035,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0036","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152036,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0037","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 7], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 7], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152037,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0038","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152038,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0039","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 8], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 8], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152039,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0040","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152040,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0041","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 9], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 9], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152041,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0042","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152042,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0043","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 10], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 10], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152043,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0044","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152044,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-subgroup-audit","semantic_template":"qf-mod46-center-subgroup-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0045","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12],[34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 11], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 11], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152045,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12],[34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,16,15,14,13,12,11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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0046","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10],[33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11],[34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12],[35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152046,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10],[33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11],[34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12],[35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0047","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11],[35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12],[36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 12], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 12], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152047,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11],[35,34,33,32,31,30,29,28,27,26,25,24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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0048","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12],[37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152048,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,8,7,6,5,4,3,2,1,0,24,23,22,21,20,1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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0049","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 13], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0, 13], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152049,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0050","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-subgroup-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25],[52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26],[53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)","explanation_zh":"Wave6第九層核對=([0], 是, 是, 是, 是, center_is_a_subgroup_by_complete_identity_product_inverse_audit)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-subgroup-audit","seed":202611152050,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,28,27,53,52,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,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25],[52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26],[53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0051","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152051,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0052","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [[0, [0, 2], 是], [1, [0, 2], 是], [2, [0, 2], 是], [3, [0, 2], 是], [4, [0, 2], 是], [5, [0, 2], 是], [6, [0, 2], 是], [7, [0, 2], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 2], [[0, [0, 2], 是], [1, [0, 2], 是], [2, [0, 2], 是], [3, [0, 2], 是], [4, [0, 2], 是], [5, [0, 2], 是], [6, [0, 2], 是], [7, [0, 2], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152052,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0053","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152053,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0054","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [[0, [0, 3], 是], [1, [0, 3], 是], [2, [0, 3], 是], [3, [0, 3], 是], [4, [0, 3], 是], [5, [0, 3], 是], [6, [0, 3], 是], [7, [0, 3], 是], [8, [0, 3], 是], [9, [0, 3], 是], [10, [0, 3], 是], [11, [0, 3], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 3], [[0, [0, 3], 是], [1, [0, 3], 是], [2, [0, 3], 是], [3, [0, 3], 是], [4, [0, 3], 是], [5, [0, 3], 是], [6, [0, 3], 是], [7, [0, 3], 是], [8, [0, 3], 是], [9, [0, 3], 是], [10, [0, 3], 是], [11, [0, 3], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152054,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0055","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152055,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0056","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [[0, [0, 4], 是], [1, [0, 4], 是], [2, [0, 4], 是], [3, [0, 4], 是], [4, [0, 4], 是], [5, [0, 4], 是], [6, [0, 4], 是], [7, [0, 4], 是], [8, [0, 4], 是], [9, [0, 4], 是], [10, [0, 4], 是], [11, [0, 4], 是], [12, [0, 4], 是], [13, [0, 4], 是], [14, [0, 4], 是], [15, [0, 4], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 4], [[0, [0, 4], 是], [1, [0, 4], 是], [2, [0, 4], 是], [3, [0, 4], 是], [4, [0, 4], 是], [5, [0, 4], 是], [6, [0, 4], 是], [7, [0, 4], 是], [8, [0, 4], 是], [9, [0, 4], 是], [10, [0, 4], 是], [11, [0, 4], 是], [12, [0, 4], 是], [13, [0, 4], 是], [14, [0, 4], 是], [15, [0, 4], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152056,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0057","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152057,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0058","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 5], [[0, [0, 5], 是], [1, [0, 5], 是], [2, [0, 5], 是], [3, [0, 5], 是], [4, [0, 5], 是], [5, [0, 5], 是], [6, [0, 5], 是], [7, [0, 5], 是], [8, [0, 5], 是], [9, [0, 5], 是], [10, [0, 5], 是], [11, [0, 5], 是], [12, [0, 5], 是], [13, [0, 5], 是], [14, [0, 5], 是], [15, [0, 5], 是], [16, [0, 5], 是], [17, [0, 5], 是], [18, [0, 5], 是], [19, [0, 5], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 5], [[0, [0, 5], 是], [1, [0, 5], 是], [2, [0, 5], 是], [3, [0, 5], 是], [4, [0, 5], 是], [5, [0, 5], 是], [6, [0, 5], 是], [7, [0, 5], 是], [8, [0, 5], 是], [9, [0, 5], 是], [10, [0, 5], 是], [11, [0, 5], 是], [12, [0, 5], 是], [13, [0, 5], 是], [14, [0, 5], 是], [15, [0, 5], 是], [16, [0, 5], 是], [17, [0, 5], 是], [18, [0, 5], 是], [19, [0, 5], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152058,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0059","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152059,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0060","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 6], [[0, [0, 6], 是], [1, [0, 6], 是], [2, [0, 6], 是], [3, [0, 6], 是], [4, [0, 6], 是], [5, [0, 6], 是], [6, [0, 6], 是], [7, [0, 6], 是], [8, [0, 6], 是], [9, [0, 6], 是], [10, [0, 6], 是], [11, [0, 6], 是], [12, [0, 6], 是], [13, [0, 6], 是], [14, [0, 6], 是], [15, [0, 6], 是], [16, [0, 6], 是], [17, [0, 6], 是], [18, [0, 6], 是], [19, [0, 6], 是], [20, [0, 6], 是], [21, [0, 6], 是], [22, [0, 6], 是], [23, [0, 6], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 6], [[0, [0, 6], 是], [1, [0, 6], 是], [2, [0, 6], 是], [3, [0, 6], 是], [4, [0, 6], 是], [5, [0, 6], 是], [6, [0, 6], 是], [7, [0, 6], 是], [8, [0, 6], 是], [9, [0, 6], 是], [10, [0, 6], 是], [11, [0, 6], 是], [12, [0, 6], 是], [13, [0, 6], 是], [14, [0, 6], 是], [15, [0, 6], 是], [16, [0, 6], 是], [17, [0, 6], 是], [18, [0, 6], 是], [19, [0, 6], 是], [20, [0, 6], 是], [21, [0, 6], 是], [22, [0, 6], 是], [23, [0, 6], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152060,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0061","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152061,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0062","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 7], [[0, [0, 7], 是], [1, [0, 7], 是], [2, [0, 7], 是], [3, [0, 7], 是], [4, [0, 7], 是], [5, [0, 7], 是], [6, [0, 7], 是], [7, [0, 7], 是], [8, [0, 7], 是], [9, [0, 7], 是], [10, [0, 7], 是], [11, [0, 7], 是], [12, [0, 7], 是], [13, [0, 7], 是], [14, [0, 7], 是], [15, [0, 7], 是], [16, [0, 7], 是], [17, [0, 7], 是], [18, [0, 7], 是], [19, [0, 7], 是], [20, [0, 7], 是], [21, [0, 7], 是], [22, [0, 7], 是], [23, [0, 7], 是], [24, [0, 7], 是], [25, [0, 7], 是], [26, [0, 7], 是], [27, [0, 7], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 7], [[0, [0, 7], 是], [1, [0, 7], 是], [2, [0, 7], 是], [3, [0, 7], 是], [4, [0, 7], 是], [5, [0, 7], 是], [6, [0, 7], 是], [7, [0, 7], 是], [8, [0, 7], 是], [9, [0, 7], 是], [10, [0, 7], 是], [11, [0, 7], 是], [12, [0, 7], 是], [13, [0, 7], 是], [14, [0, 7], 是], [15, [0, 7], 是], [16, [0, 7], 是], [17, [0, 7], 是], [18, [0, 7], 是], [19, [0, 7], 是], [20, [0, 7], 是], [21, [0, 7], 是], [22, [0, 7], 是], [23, [0, 7], 是], [24, [0, 7], 是], [25, [0, 7], 是], [26, [0, 7], 是], [27, [0, 7], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152062,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0063","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152063,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0064","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 8], [[0, [0, 8], 是], [1, [0, 8], 是], [2, [0, 8], 是], [3, [0, 8], 是], [4, [0, 8], 是], [5, [0, 8], 是], [6, [0, 8], 是], [7, [0, 8], 是], [8, [0, 8], 是], [9, [0, 8], 是], [10, [0, 8], 是], [11, [0, 8], 是], [12, [0, 8], 是], [13, [0, 8], 是], [14, [0, 8], 是], [15, [0, 8], 是], [16, [0, 8], 是], [17, [0, 8], 是], [18, [0, 8], 是], [19, [0, 8], 是], [20, [0, 8], 是], [21, [0, 8], 是], [22, [0, 8], 是], [23, [0, 8], 是], [24, [0, 8], 是], [25, [0, 8], 是], [26, [0, 8], 是], [27, [0, 8], 是], [28, [0, 8], 是], [29, [0, 8], 是], [30, [0, 8], 是], [31, [0, 8], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 8], [[0, [0, 8], 是], [1, [0, 8], 是], [2, [0, 8], 是], [3, [0, 8], 是], [4, [0, 8], 是], [5, [0, 8], 是], [6, [0, 8], 是], [7, [0, 8], 是], [8, [0, 8], 是], [9, [0, 8], 是], [10, [0, 8], 是], [11, [0, 8], 是], [12, [0, 8], 是], [13, [0, 8], 是], [14, [0, 8], 是], [15, [0, 8], 是], [16, [0, 8], 是], [17, [0, 8], 是], [18, [0, 8], 是], [19, [0, 8], 是], [20, [0, 8], 是], [21, [0, 8], 是], [22, [0, 8], 是], [23, [0, 8], 是], [24, [0, 8], 是], [25, [0, 8], 是], [26, [0, 8], 是], [27, [0, 8], 是], [28, [0, 8], 是], [29, [0, 8], 是], [30, [0, 8], 是], [31, [0, 8], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152064,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0065","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152065,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0066","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 9], [[0, [0, 9], 是], [1, [0, 9], 是], [2, [0, 9], 是], [3, [0, 9], 是], [4, [0, 9], 是], [5, [0, 9], 是], [6, [0, 9], 是], [7, [0, 9], 是], [8, [0, 9], 是], [9, [0, 9], 是], [10, [0, 9], 是], [11, [0, 9], 是], [12, [0, 9], 是], [13, [0, 9], 是], [14, [0, 9], 是], [15, [0, 9], 是], [16, [0, 9], 是], [17, [0, 9], 是], [18, [0, 9], 是], [19, [0, 9], 是], [20, [0, 9], 是], [21, [0, 9], 是], [22, [0, 9], 是], [23, [0, 9], 是], [24, [0, 9], 是], [25, [0, 9], 是], [26, [0, 9], 是], [27, [0, 9], 是], [28, [0, 9], 是], [29, [0, 9], 是], [30, [0, 9], 是], [31, [0, 9], 是], [32, [0, 9], 是], [33, [0, 9], 是], [34, [0, 9], 是], [35, [0, 9], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 9], [[0, [0, 9], 是], [1, [0, 9], 是], [2, [0, 9], 是], [3, [0, 9], 是], [4, [0, 9], 是], [5, [0, 9], 是], [6, [0, 9], 是], [7, [0, 9], 是], [8, [0, 9], 是], [9, [0, 9], 是], [10, [0, 9], 是], [11, [0, 9], 是], [12, [0, 9], 是], [13, [0, 9], 是], [14, [0, 9], 是], [15, [0, 9], 是], [16, [0, 9], 是], [17, [0, 9], 是], [18, [0, 9], 是], [19, [0, 9], 是], [20, [0, 9], 是], [21, [0, 9], 是], [22, [0, 9], 是], [23, [0, 9], 是], [24, [0, 9], 是], [25, [0, 9], 是], [26, [0, 9], 是], [27, [0, 9], 是], [28, [0, 9], 是], [29, [0, 9], 是], [30, [0, 9], 是], [31, [0, 9], 是], [32, [0, 9], 是], [33, [0, 9], 是], [34, [0, 9], 是], [35, [0, 9], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152066,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0067","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152067,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0068","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 10], [[0, [0, 10], 是], [1, [0, 10], 是], [2, [0, 10], 是], [3, [0, 10], 是], [4, [0, 10], 是], [5, [0, 10], 是], [6, [0, 10], 是], [7, [0, 10], 是], [8, [0, 10], 是], [9, [0, 10], 是], [10, [0, 10], 是], [11, [0, 10], 是], [12, [0, 10], 是], [13, [0, 10], 是], [14, [0, 10], 是], [15, [0, 10], 是], [16, [0, 10], 是], [17, [0, 10], 是], [18, [0, 10], 是], [19, [0, 10], 是], [20, [0, 10], 是], [21, [0, 10], 是], [22, [0, 10], 是], [23, [0, 10], 是], [24, [0, 10], 是], [25, [0, 10], 是], [26, [0, 10], 是], [27, [0, 10], 是], [28, [0, 10], 是], [29, [0, 10], 是], [30, [0, 10], 是], [31, [0, 10], 是], [32, [0, 10], 是], [33, [0, 10], 是], [34, [0, 10], 是], [35, [0, 10], 是], [36, [0, 10], 是], [37, [0, 10], 是], [38, [0, 10], 是], [39, [0, 10], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 10], [[0, [0, 10], 是], [1, [0, 10], 是], [2, [0, 10], 是], [3, [0, 10], 是], [4, [0, 10], 是], [5, [0, 10], 是], [6, [0, 10], 是], [7, [0, 10], 是], [8, [0, 10], 是], [9, [0, 10], 是], [10, [0, 10], 是], [11, [0, 10], 是], [12, [0, 10], 是], [13, [0, 10], 是], [14, [0, 10], 是], [15, [0, 10], 是], [16, [0, 10], 是], [17, [0, 10], 是], [18, [0, 10], 是], [19, [0, 10], 是], [20, [0, 10], 是], [21, [0, 10], 是], [22, [0, 10], 是], [23, [0, 10], 是], [24, [0, 10], 是], [25, [0, 10], 是], [26, [0, 10], 是], [27, [0, 10], 是], [28, [0, 10], 是], [29, [0, 10], 是], [30, [0, 10], 是], [31, [0, 10], 是], [32, [0, 10], 是], [33, [0, 10], 是], [34, [0, 10], 是], [35, [0, 10], 是], [36, [0, 10], 是], [37, [0, 10], 是], [38, [0, 10], 是], [39, [0, 10], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152068,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"domain":"Complete_finite_group_center","operation":"center-normality-audit","semantic_template":"qf-mod46-center-normality-audit/{\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0069","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是], [38, [0], 是], [39, [0], 是], [40, [0], 是], [41, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是], [38, [0], 是], [39, [0], 是], [40, [0], 是], [41, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152069,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0070","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12],[34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 11], [[0, [0, 11], 是], [1, [0, 11], 是], [2, [0, 11], 是], [3, [0, 11], 是], [4, [0, 11], 是], [5, [0, 11], 是], [6, [0, 11], 是], [7, [0, 11], 是], [8, [0, 11], 是], [9, [0, 11], 是], [10, [0, 11], 是], [11, [0, 11], 是], [12, [0, 11], 是], [13, [0, 11], 是], [14, [0, 11], 是], [15, [0, 11], 是], [16, [0, 11], 是], [17, [0, 11], 是], [18, [0, 11], 是], [19, [0, 11], 是], [20, [0, 11], 是], [21, [0, 11], 是], [22, [0, 11], 是], [23, [0, 11], 是], [24, [0, 11], 是], [25, [0, 11], 是], [26, [0, 11], 是], [27, [0, 11], 是], [28, [0, 11], 是], [29, [0, 11], 是], [30, [0, 11], 是], [31, [0, 11], 是], [32, [0, 11], 是], [33, [0, 11], 是], [34, [0, 11], 是], [35, [0, 11], 是], [36, [0, 11], 是], [37, [0, 11], 是], [38, [0, 11], 是], [39, [0, 11], 是], [40, [0, 11], 是], [41, [0, 11], 是], [42, [0, 11], 是], [43, [0, 11], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 11], [[0, [0, 11], 是], [1, [0, 11], 是], [2, [0, 11], 是], [3, [0, 11], 是], [4, [0, 11], 是], [5, [0, 11], 是], [6, [0, 11], 是], [7, [0, 11], 是], [8, [0, 11], 是], [9, [0, 11], 是], [10, [0, 11], 是], [11, [0, 11], 是], [12, [0, 11], 是], [13, [0, 11], 是], [14, [0, 11], 是], [15, [0, 11], 是], [16, [0, 11], 是], [17, [0, 11], 是], [18, [0, 11], 是], [19, [0, 11], 是], [20, [0, 11], 是], [21, [0, 11], 是], [22, [0, 11], 是], [23, [0, 11], 是], [24, [0, 11], 是], [25, [0, 11], 是], [26, [0, 11], 是], [27, [0, 11], 是], [28, [0, 11], 是], [29, [0, 11], 是], [30, [0, 11], 是], [31, [0, 11], 是], [32, [0, 11], 是], [33, [0, 11], 是], [34, [0, 11], 是], [35, [0, 11], 是], [36, [0, 11], 是], [37, [0, 11], 是], [38, [0, 11], 是], [39, [0, 11], 是], [40, [0, 11], 是], [41, [0, 11], 是], [42, [0, 11], 是], [43, [0, 11], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152070,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12],[34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,16,15,14,13,12,1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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0071","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10],[33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11],[34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12],[35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是], [38, [0], 是], [39, [0], 是], [40, [0], 是], [41, [0], 是], [42, [0], 是], [43, [0], 是], [44, [0], 是], [45, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是], [38, [0], 是], [39, [0], 是], [40, [0], 是], [41, [0], 是], [42, [0], 是], [43, [0], 是], [44, [0], 是], [45, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152071,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10],[33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11],[34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12],[35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,1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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0072","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11],[35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12],[36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 12], [[0, [0, 12], 是], [1, [0, 12], 是], [2, [0, 12], 是], [3, [0, 12], 是], [4, [0, 12], 是], [5, [0, 12], 是], [6, [0, 12], 是], [7, [0, 12], 是], [8, [0, 12], 是], [9, [0, 12], 是], [10, [0, 12], 是], [11, [0, 12], 是], [12, [0, 12], 是], [13, [0, 12], 是], [14, [0, 12], 是], [15, [0, 12], 是], [16, [0, 12], 是], [17, [0, 12], 是], [18, [0, 12], 是], [19, [0, 12], 是], [20, [0, 12], 是], [21, [0, 12], 是], [22, [0, 12], 是], [23, [0, 12], 是], [24, [0, 12], 是], [25, [0, 12], 是], [26, [0, 12], 是], [27, [0, 12], 是], [28, [0, 12], 是], [29, [0, 12], 是], [30, [0, 12], 是], [31, [0, 12], 是], [32, [0, 12], 是], [33, [0, 12], 是], [34, [0, 12], 是], [35, [0, 12], 是], [36, [0, 12], 是], [37, [0, 12], 是], [38, [0, 12], 是], [39, [0, 12], 是], [40, [0, 12], 是], [41, [0, 12], 是], [42, [0, 12], 是], [43, [0, 12], 是], [44, [0, 12], 是], [45, [0, 12], 是], [46, [0, 12], 是], [47, [0, 12], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 12], [[0, [0, 12], 是], [1, [0, 12], 是], [2, [0, 12], 是], [3, [0, 12], 是], [4, [0, 12], 是], [5, [0, 12], 是], [6, [0, 12], 是], [7, [0, 12], 是], [8, [0, 12], 是], [9, [0, 12], 是], [10, [0, 12], 是], [11, [0, 12], 是], [12, [0, 12], 是], [13, [0, 12], 是], [14, [0, 12], 是], [15, [0, 12], 是], [16, [0, 12], 是], [17, [0, 12], 是], [18, [0, 12], 是], [19, [0, 12], 是], [20, [0, 12], 是], [21, [0, 12], 是], [22, [0, 12], 是], [23, [0, 12], 是], [24, [0, 12], 是], [25, [0, 12], 是], [26, [0, 12], 是], [27, [0, 12], 是], [28, [0, 12], 是], [29, [0, 12], 是], [30, [0, 12], 是], [31, [0, 12], 是], [32, [0, 12], 是], [33, [0, 12], 是], [34, [0, 12], 是], [35, [0, 12], 是], [36, [0, 12], 是], [37, [0, 12], 是], [38, [0, 12], 是], [39, [0, 12], 是], [40, [0, 12], 是], [41, [0, 12], 是], [42, [0, 12], 是], [43, [0, 12], 是], [44, [0, 12], 是], [45, [0, 12], 是], [46, [0, 12], 是], [47, [0, 12], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152072,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11],[35,34,33,32,31,30,29,28,27,26,25,2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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0073","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12],[37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是], [38, [0], 是], [39, [0], 是], [40, [0], 是], [41, [0], 是], [42, [0], 是], [43, [0], 是], [44, [0], 是], [45, [0], 是], [46, [0], 是], [47, [0], 是], [48, [0], 是], [49, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是], [38, [0], 是], [39, [0], 是], [40, [0], 是], [41, [0], 是], [42, [0], 是], [43, [0], 是], [44, [0], 是], [45, [0], 是], [46, [0], 是], [47, [0], 是], [48, [0], 是], [49, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152073,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,8,7,6,5,4,3,2,1,0,24,23,22,21,20,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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0074","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 13], [[0, [0, 13], 是], [1, [0, 13], 是], [2, [0, 13], 是], [3, [0, 13], 是], [4, [0, 13], 是], [5, [0, 13], 是], [6, [0, 13], 是], [7, [0, 13], 是], [8, [0, 13], 是], [9, [0, 13], 是], [10, [0, 13], 是], [11, [0, 13], 是], [12, [0, 13], 是], [13, [0, 13], 是], [14, [0, 13], 是], [15, [0, 13], 是], [16, [0, 13], 是], [17, [0, 13], 是], [18, [0, 13], 是], [19, [0, 13], 是], [20, [0, 13], 是], [21, [0, 13], 是], [22, [0, 13], 是], [23, [0, 13], 是], [24, [0, 13], 是], [25, [0, 13], 是], [26, [0, 13], 是], [27, [0, 13], 是], [28, [0, 13], 是], [29, [0, 13], 是], [30, [0, 13], 是], [31, [0, 13], 是], [32, [0, 13], 是], [33, [0, 13], 是], [34, [0, 13], 是], [35, [0, 13], 是], [36, [0, 13], 是], [37, [0, 13], 是], [38, [0, 13], 是], [39, [0, 13], 是], [40, [0, 13], 是], [41, [0, 13], 是], [42, [0, 13], 是], [43, [0, 13], 是], [44, [0, 13], 是], [45, [0, 13], 是], [46, [0, 13], 是], [47, [0, 13], 是], [48, [0, 13], 是], [49, [0, 13], 是], [50, [0, 13], 是], [51, [0, 13], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0, 13], [[0, [0, 13], 是], [1, [0, 13], 是], [2, [0, 13], 是], [3, [0, 13], 是], [4, [0, 13], 是], [5, [0, 13], 是], [6, [0, 13], 是], [7, [0, 13], 是], [8, [0, 13], 是], [9, [0, 13], 是], [10, [0, 13], 是], [11, [0, 13], 是], [12, [0, 13], 是], [13, [0, 13], 是], [14, [0, 13], 是], [15, [0, 13], 是], [16, [0, 13], 是], [17, [0, 13], 是], [18, [0, 13], 是], [19, [0, 13], 是], [20, [0, 13], 是], [21, [0, 13], 是], [22, [0, 13], 是], [23, [0, 13], 是], [24, [0, 13], 是], [25, [0, 13], 是], [26, [0, 13], 是], [27, [0, 13], 是], [28, [0, 13], 是], [29, [0, 13], 是], [30, [0, 13], 是], [31, [0, 13], 是], [32, [0, 13], 是], [33, [0, 13], 是], [34, [0, 13], 是], [35, [0, 13], 是], [36, [0, 13], 是], [37, [0, 13], 是], [38, [0, 13], 是], [39, [0, 13], 是], [40, [0, 13], 是], [41, [0, 13], 是], [42, [0, 13], 是], [43, [0, 13], 是], [44, [0, 13], 是], [45, [0, 13], 是], [46, [0, 13], 是], [47, [0, 13], 是], [48, [0, 13], 是], [49, [0, 13], 是], [50, [0, 13], 是], [51, [0, 13], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152074,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,3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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0075","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-normality-audit；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25],[52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26],[53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是], [38, [0], 是], [39, [0], 是], [40, [0], 是], [41, [0], 是], [42, [0], 是], [43, [0], 是], [44, [0], 是], [45, [0], 是], [46, [0], 是], [47, [0], 是], [48, [0], 是], [49, [0], 是], [50, [0], 是], [51, [0], 是], [52, [0], 是], [53, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)","explanation_zh":"Wave6第九層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是], [12, [0], 是], [13, [0], 是], [14, [0], 是], [15, [0], 是], [16, [0], 是], [17, [0], 是], [18, [0], 是], [19, [0], 是], [20, [0], 是], [21, [0], 是], [22, [0], 是], [23, [0], 是], [24, [0], 是], [25, [0], 是], [26, [0], 是], [27, [0], 是], [28, [0], 是], [29, [0], 是], [30, [0], 是], [31, [0], 是], [32, [0], 是], [33, [0], 是], [34, [0], 是], [35, [0], 是], [36, [0], 是], [37, [0], 是], [38, [0], 是], [39, [0], 是], [40, [0], 是], [41, [0], 是], [42, [0], 是], [43, [0], 是], [44, [0], 是], [45, [0], 是], [46, [0], 是], [47, [0], 是], [48, [0], 是], [49, [0], 是], [50, [0], 是], [51, [0], 是], [52, [0], 是], [53, [0], 是]], 是, every_conjugate_of_a_central_element_remains_the_same_central_element)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-normality-audit","seed":202611152075,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,28,27,53,52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9,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25],[52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26],[53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0076","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],\"identity\":0,\"element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 否, [3, 4, 5], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(3, 否, [3, 4, 5], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152076,"parameters":{"group_table":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],"identity":0,"element":3,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":3,\"group_table\":[[0,1,2,3,4,5],[1,2,0,4,5,3],[2,0,1,5,3,4],[3,5,4,0,2,1],[4,3,5,1,0,2],[5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0077","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 否, [4, 6], 否, 是, [0, 2], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(4, 否, [4, 6], 否, 是, [0, 2], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152077,"parameters":{"group_table":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],"identity":0,"element":4,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":4,\"group_table\":[[0,1,2,3,4,5,6,7],[1,2,3,0,5,6,7,4],[2,3,0,1,6,7,4,5],[3,0,1,2,7,4,5,6],[4,7,6,5,0,3,2,1],[5,4,7,6,1,0,3,2],[6,5,4,7,2,1,0,3],[7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0078","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 否, [5, 6, 7, 8, 9], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(5, 否, [5, 6, 7, 8, 9], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152078,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":5,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":5,\"group_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,0,6,7,8,9,5],[2,3,4,0,1,7,8,9,5,6],[3,4,0,1,2,8,9,5,6,7],[4,0,1,2,3,9,5,6,7,8],[5,9,8,7,6,0,4,3,2,1],[6,5,9,8,7,1,0,4,3,2],[7,6,5,9,8,2,1,0,4,3],[8,7,6,5,9,3,2,1,0,4],[9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0079","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, [3], 是, 是, [0, 3], 否, 否, 是, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(3, 是, [3], 是, 是, [0, 3], 否, 否, 是, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152079,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":3,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":3,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,0,7,8,9,10,11,6],[2,3,4,5,0,1,8,9,10,11,6,7],[3,4,5,0,1,2,9,10,11,6,7,8],[4,5,0,1,2,3,10,11,6,7,8,9],[5,0,1,2,3,4,11,6,7,8,9,10],[6,11,10,9,8,7,0,5,4,3,2,1],[7,6,11,10,9,8,1,0,5,4,3,2],[8,7,6,11,10,9,2,1,0,5,4,3],[9,8,7,6,11,10,3,2,1,0,5,4],[10,9,8,7,6,11,4,3,2,1,0,5],[11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0080","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 否, [7, 8, 9, 10, 11, 12, 13], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(7, 否, [7, 8, 9, 10, 11, 12, 13], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152080,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":7,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":7,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,0,8,9,10,11,12,13,7],[2,3,4,5,6,0,1,9,10,11,12,13,7,8],[3,4,5,6,0,1,2,10,11,12,13,7,8,9],[4,5,6,0,1,2,3,11,12,13,7,8,9,10],[5,6,0,1,2,3,4,12,13,7,8,9,10,11],[6,0,1,2,3,4,5,13,7,8,9,10,11,12],[7,13,12,11,10,9,8,0,6,5,4,3,2,1],[8,7,13,12,11,10,9,1,0,6,5,4,3,2],[9,8,7,13,12,11,10,2,1,0,6,5,4,3],[10,9,8,7,13,12,11,3,2,1,0,6,5,4],[11,10,9,8,7,13,12,4,3,2,1,0,6,5],[12,11,10,9,8,7,13,5,4,3,2,1,0,6],[13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0081","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 否, [8, 10, 12, 14], 否, 是, [0, 4], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(8, 否, [8, 10, 12, 14], 否, 是, [0, 4], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152081,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":8,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":8,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,0,9,10,11,12,13,14,15,8],[2,3,4,5,6,7,0,1,10,11,12,13,14,15,8,9],[3,4,5,6,7,0,1,2,11,12,13,14,15,8,9,10],[4,5,6,7,0,1,2,3,12,13,14,15,8,9,10,11],[5,6,7,0,1,2,3,4,13,14,15,8,9,10,11,12],[6,7,0,1,2,3,4,5,14,15,8,9,10,11,12,13],[7,0,1,2,3,4,5,6,15,8,9,10,11,12,13,14],[8,15,14,13,12,11,10,9,0,7,6,5,4,3,2,1],[9,8,15,14,13,12,11,10,1,0,7,6,5,4,3,2],[10,9,8,15,14,13,12,11,2,1,0,7,6,5,4,3],[11,10,9,8,15,14,13,12,3,2,1,0,7,6,5,4],[12,11,10,9,8,15,14,13,4,3,2,1,0,7,6,5],[13,12,11,10,9,8,15,14,5,4,3,2,1,0,7,6],[14,13,12,11,10,9,8,15,6,5,4,3,2,1,0,7],[15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0082","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 否, [9, 10, 11, 12, 13, 14, 15, 16, 17], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(9, 否, [9, 10, 11, 12, 13, 14, 15, 16, 17], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152082,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":9,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":9,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0083","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 否, [10, 12, 14, 16, 18], 否, 是, [0, 5], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(10, 否, [10, 12, 14, 16, 18], 否, 是, [0, 5], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152083,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":10,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":10,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,0,11,12,13,14,15,16,17,18,19,10],[2,3,4,5,6,7,8,9,0,1,12,13,14,15,16,17,18,19,10,11],[3,4,5,6,7,8,9,0,1,2,13,14,15,16,17,18,19,10,11,12],[4,5,6,7,8,9,0,1,2,3,14,15,16,17,18,19,10,11,12,13],[5,6,7,8,9,0,1,2,3,4,15,16,17,18,19,10,11,12,13,14],[6,7,8,9,0,1,2,3,4,5,16,17,18,19,10,11,12,13,14,15],[7,8,9,0,1,2,3,4,5,6,17,18,19,10,11,12,13,14,15,16],[8,9,0,1,2,3,4,5,6,7,18,19,10,11,12,13,14,15,16,17],[9,0,1,2,3,4,5,6,7,8,19,10,11,12,13,14,15,16,17,18],[10,19,18,17,16,15,14,13,12,11,0,9,8,7,6,5,4,3,2,1],[11,10,19,18,17,16,15,14,13,12,1,0,9,8,7,6,5,4,3,2],[12,11,10,19,18,17,16,15,14,13,2,1,0,9,8,7,6,5,4,3],[13,12,11,10,19,18,17,16,15,14,3,2,1,0,9,8,7,6,5,4],[14,13,12,11,10,19,18,17,16,15,4,3,2,1,0,9,8,7,6,5],[15,14,13,12,11,10,19,18,17,16,5,4,3,2,1,0,9,8,7,6],[16,15,14,13,12,11,10,19,18,17,6,5,4,3,2,1,0,9,8,7],[17,16,15,14,13,12,11,10,19,18,7,6,5,4,3,2,1,0,9,8],[18,17,16,15,14,13,12,11,10,19,8,7,6,5,4,3,2,1,0,9],[19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0084","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 否, [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(11, 否, [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152084,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":11,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":11,\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,0,12,13,14,15,16,17,18,19,20,21,11],[2,3,4,5,6,7,8,9,10,0,1,13,14,15,16,17,18,19,20,21,11,12],[3,4,5,6,7,8,9,10,0,1,2,14,15,16,17,18,19,20,21,11,12,13],[4,5,6,7,8,9,10,0,1,2,3,15,16,17,18,19,20,21,11,12,13,14],[5,6,7,8,9,10,0,1,2,3,4,16,17,18,19,20,21,11,12,13,14,15],[6,7,8,9,10,0,1,2,3,4,5,17,18,19,20,21,11,12,13,14,15,16],[7,8,9,10,0,1,2,3,4,5,6,18,19,20,21,11,12,13,14,15,16,17],[8,9,10,0,1,2,3,4,5,6,7,19,20,21,11,12,13,14,15,16,17,18],[9,10,0,1,2,3,4,5,6,7,8,20,21,11,12,13,14,15,16,17,18,19],[10,0,1,2,3,4,5,6,7,8,9,21,11,12,13,14,15,16,17,18,19,20],[11,21,20,19,18,17,16,15,14,13,12,0,10,9,8,7,6,5,4,3,2,1],[12,11,21,20,19,18,17,16,15,14,13,1,0,10,9,8,7,6,5,4,3,2],[13,12,11,21,20,19,18,17,16,15,14,2,1,0,10,9,8,7,6,5,4,3],[14,13,12,11,21,20,19,18,17,16,15,3,2,1,0,10,9,8,7,6,5,4],[15,14,13,12,11,21,20,19,18,17,16,4,3,2,1,0,10,9,8,7,6,5],[16,15,14,13,12,11,21,20,19,18,17,5,4,3,2,1,0,10,9,8,7,6],[17,16,15,14,13,12,11,21,20,19,18,6,5,4,3,2,1,0,10,9,8,7],[18,17,16,15,14,13,12,11,21,20,19,7,6,5,4,3,2,1,0,10,9,8],[19,18,17,16,15,14,13,12,11,21,20,8,7,6,5,4,3,2,1,0,10,9],[20,19,18,17,16,15,14,13,12,11,21,9,8,7,6,5,4,3,2,1,0,10],[21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0085","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 是, [6], 是, 是, [0, 6], 否, 否, 是, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(6, 是, [6], 是, 是, [0, 6], 否, 否, 是, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152085,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":6,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":6,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,0,13,14,15,16,17,18,19,20,21,22,23,12],[2,3,4,5,6,7,8,9,10,11,0,1,14,15,16,17,18,19,20,21,22,23,12,13],[3,4,5,6,7,8,9,10,11,0,1,2,15,16,17,18,19,20,21,22,23,12,13,14],[4,5,6,7,8,9,10,11,0,1,2,3,16,17,18,19,20,21,22,23,12,13,14,15],[5,6,7,8,9,10,11,0,1,2,3,4,17,18,19,20,21,22,23,12,13,14,15,16],[6,7,8,9,10,11,0,1,2,3,4,5,18,19,20,21,22,23,12,13,14,15,16,17],[7,8,9,10,11,0,1,2,3,4,5,6,19,20,21,22,23,12,13,14,15,16,17,18],[8,9,10,11,0,1,2,3,4,5,6,7,20,21,22,23,12,13,14,15,16,17,18,19],[9,10,11,0,1,2,3,4,5,6,7,8,21,22,23,12,13,14,15,16,17,18,19,20],[10,11,0,1,2,3,4,5,6,7,8,9,22,23,12,13,14,15,16,17,18,19,20,21],[11,0,1,2,3,4,5,6,7,8,9,10,23,12,13,14,15,16,17,18,19,20,21,22],[12,23,22,21,20,19,18,17,16,15,14,13,0,11,10,9,8,7,6,5,4,3,2,1],[13,12,23,22,21,20,19,18,17,16,15,14,1,0,11,10,9,8,7,6,5,4,3,2],[14,13,12,23,22,21,20,19,18,17,16,15,2,1,0,11,10,9,8,7,6,5,4,3],[15,14,13,12,23,22,21,20,19,18,17,16,3,2,1,0,11,10,9,8,7,6,5,4],[16,15,14,13,12,23,22,21,20,19,18,17,4,3,2,1,0,11,10,9,8,7,6,5],[17,16,15,14,13,12,23,22,21,20,19,18,5,4,3,2,1,0,11,10,9,8,7,6],[18,17,16,15,14,13,12,23,22,21,20,19,6,5,4,3,2,1,0,11,10,9,8,7],[19,18,17,16,15,14,13,12,23,22,21,20,7,6,5,4,3,2,1,0,11,10,9,8],[20,19,18,17,16,15,14,13,12,23,22,21,8,7,6,5,4,3,2,1,0,11,10,9],[21,20,19,18,17,16,15,14,13,12,23,22,9,8,7,6,5,4,3,2,1,0,11,10],[22,21,20,19,18,17,16,15,14,13,12,23,10,9,8,7,6,5,4,3,2,1,0,11],[23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0086","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],\"identity\":0,\"element\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 否, [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(13, 否, [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152086,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],"identity":0,"element":13,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":13,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,0,14,15,16,17,18,19,20,21,22,23,24,25,13],[2,3,4,5,6,7,8,9,10,11,12,0,1,15,16,17,18,19,20,21,22,23,24,25,13,14],[3,4,5,6,7,8,9,10,11,12,0,1,2,16,17,18,19,20,21,22,23,24,25,13,14,15],[4,5,6,7,8,9,10,11,12,0,1,2,3,17,18,19,20,21,22,23,24,25,13,14,15,16],[5,6,7,8,9,10,11,12,0,1,2,3,4,18,19,20,21,22,23,24,25,13,14,15,16,17],[6,7,8,9,10,11,12,0,1,2,3,4,5,19,20,21,22,23,24,25,13,14,15,16,17,18],[7,8,9,10,11,12,0,1,2,3,4,5,6,20,21,22,23,24,25,13,14,15,16,17,18,19],[8,9,10,11,12,0,1,2,3,4,5,6,7,21,22,23,24,25,13,14,15,16,17,18,19,20],[9,10,11,12,0,1,2,3,4,5,6,7,8,22,23,24,25,13,14,15,16,17,18,19,20,21],[10,11,12,0,1,2,3,4,5,6,7,8,9,23,24,25,13,14,15,16,17,18,19,20,21,22],[11,12,0,1,2,3,4,5,6,7,8,9,10,24,25,13,14,15,16,17,18,19,20,21,22,23],[12,0,1,2,3,4,5,6,7,8,9,10,11,25,13,14,15,16,17,18,19,20,21,22,23,24],[13,25,24,23,22,21,20,19,18,17,16,15,14,0,12,11,10,9,8,7,6,5,4,3,2,1],[14,13,25,24,23,22,21,20,19,18,17,16,15,1,0,12,11,10,9,8,7,6,5,4,3,2],[15,14,13,25,24,23,22,21,20,19,18,17,16,2,1,0,12,11,10,9,8,7,6,5,4,3],[16,15,14,13,25,24,23,22,21,20,19,18,17,3,2,1,0,12,11,10,9,8,7,6,5,4],[17,16,15,14,13,25,24,23,22,21,20,19,18,4,3,2,1,0,12,11,10,9,8,7,6,5],[18,17,16,15,14,13,25,24,23,22,21,20,19,5,4,3,2,1,0,12,11,10,9,8,7,6],[19,18,17,16,15,14,13,25,24,23,22,21,20,6,5,4,3,2,1,0,12,11,10,9,8,7],[20,19,18,17,16,15,14,13,25,24,23,22,21,7,6,5,4,3,2,1,0,12,11,10,9,8],[21,20,19,18,17,16,15,14,13,25,24,23,22,8,7,6,5,4,3,2,1,0,12,11,10,9],[22,21,20,19,18,17,16,15,14,13,25,24,23,9,8,7,6,5,4,3,2,1,0,12,11,10],[23,22,21,20,19,18,17,16,15,14,13,25,24,10,9,8,7,6,5,4,3,2,1,0,12,11],[24,23,22,21,20,19,18,17,16,15,14,13,25,11,10,9,8,7,6,5,4,3,2,1,0,12],[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]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0087","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 否, [14, 16, 18, 20, 22, 24, 26], 否, 是, [0, 7], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(14, 否, [14, 16, 18, 20, 22, 24, 26], 否, 是, [0, 7], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152087,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":14,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":14,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,0,15,16,17,18,19,20,21,22,23,24,25,26,27,14],[2,3,4,5,6,7,8,9,10,11,12,13,0,1,16,17,18,19,20,21,22,23,24,25,26,27,14,15],[3,4,5,6,7,8,9,10,11,12,13,0,1,2,17,18,19,20,21,22,23,24,25,26,27,14,15,16],[4,5,6,7,8,9,10,11,12,13,0,1,2,3,18,19,20,21,22,23,24,25,26,27,14,15,16,17],[5,6,7,8,9,10,11,12,13,0,1,2,3,4,19,20,21,22,23,24,25,26,27,14,15,16,17,18],[6,7,8,9,10,11,12,13,0,1,2,3,4,5,20,21,22,23,24,25,26,27,14,15,16,17,18,19],[7,8,9,10,11,12,13,0,1,2,3,4,5,6,21,22,23,24,25,26,27,14,15,16,17,18,19,20],[8,9,10,11,12,13,0,1,2,3,4,5,6,7,22,23,24,25,26,27,14,15,16,17,18,19,20,21],[9,10,11,12,13,0,1,2,3,4,5,6,7,8,23,24,25,26,27,14,15,16,17,18,19,20,21,22],[10,11,12,13,0,1,2,3,4,5,6,7,8,9,24,25,26,27,14,15,16,17,18,19,20,21,22,23],[11,12,13,0,1,2,3,4,5,6,7,8,9,10,25,26,27,14,15,16,17,18,19,20,21,22,23,24],[12,13,0,1,2,3,4,5,6,7,8,9,10,11,26,27,14,15,16,17,18,19,20,21,22,23,24,25],[13,0,1,2,3,4,5,6,7,8,9,10,11,12,27,14,15,16,17,18,19,20,21,22,23,24,25,26],[14,27,26,25,24,23,22,21,20,19,18,17,16,15,0,13,12,11,10,9,8,7,6,5,4,3,2,1],[15,14,27,26,25,24,23,22,21,20,19,18,17,16,1,0,13,12,11,10,9,8,7,6,5,4,3,2],[16,15,14,27,26,25,24,23,22,21,20,19,18,17,2,1,0,13,12,11,10,9,8,7,6,5,4,3],[17,16,15,14,27,26,25,24,23,22,21,20,19,18,3,2,1,0,13,12,11,10,9,8,7,6,5,4],[18,17,16,15,14,27,26,25,24,23,22,21,20,19,4,3,2,1,0,13,12,11,10,9,8,7,6,5],[19,18,17,16,15,14,27,26,25,24,23,22,21,20,5,4,3,2,1,0,13,12,11,10,9,8,7,6],[20,19,18,17,16,15,14,27,26,25,24,23,22,21,6,5,4,3,2,1,0,13,12,11,10,9,8,7],[21,20,19,18,17,16,15,14,27,26,25,24,23,22,7,6,5,4,3,2,1,0,13,12,11,10,9,8],[22,21,20,19,18,17,16,15,14,27,26,25,24,23,8,7,6,5,4,3,2,1,0,13,12,11,10,9],[23,22,21,20,19,18,17,16,15,14,27,26,25,24,9,8,7,6,5,4,3,2,1,0,13,12,11,10],[24,23,22,21,20,19,18,17,16,15,14,27,26,25,10,9,8,7,6,5,4,3,2,1,0,13,12,11],[25,24,23,22,21,20,19,18,17,16,15,14,27,26,11,10,9,8,7,6,5,4,3,2,1,0,13,12],[26,25,24,23,22,21,20,19,18,17,16,15,14,27,12,11,10,9,8,7,6,5,4,3,2,1,0,13],[27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0088","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 否, [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(15, 否, [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152088,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":15,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":15,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,16,17,18,19,20,21,22,23,24,25,26,27,28,29,15],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1,17,18,19,20,21,22,23,24,25,26,27,28,29,15,16],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2,18,19,20,21,22,23,24,25,26,27,28,29,15,16,17],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3,19,20,21,22,23,24,25,26,27,28,29,15,16,17,18],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4,20,21,22,23,24,25,26,27,28,29,15,16,17,18,19],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5,21,22,23,24,25,26,27,28,29,15,16,17,18,19,20],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6,22,23,24,25,26,27,28,29,15,16,17,18,19,20,21],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7,23,24,25,26,27,28,29,15,16,17,18,19,20,21,22],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8,24,25,26,27,28,29,15,16,17,18,19,20,21,22,23],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9,25,26,27,28,29,15,16,17,18,19,20,21,22,23,24],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10,26,27,28,29,15,16,17,18,19,20,21,22,23,24,25],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11,27,28,29,15,16,17,18,19,20,21,22,23,24,25,26],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12,28,29,15,16,17,18,19,20,21,22,23,24,25,26,27],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13,29,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[15,29,28,27,26,25,24,23,22,21,20,19,18,17,16,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[16,15,29,28,27,26,25,24,23,22,21,20,19,18,17,1,0,14,13,12,11,10,9,8,7,6,5,4,3,2],[17,16,15,29,28,27,26,25,24,23,22,21,20,19,18,2,1,0,14,13,12,11,10,9,8,7,6,5,4,3],[18,17,16,15,29,28,27,26,25,24,23,22,21,20,19,3,2,1,0,14,13,12,11,10,9,8,7,6,5,4],[19,18,17,16,15,29,28,27,26,25,24,23,22,21,20,4,3,2,1,0,14,13,12,11,10,9,8,7,6,5],[20,19,18,17,16,15,29,28,27,26,25,24,23,22,21,5,4,3,2,1,0,14,13,12,11,10,9,8,7,6],[21,20,19,18,17,16,15,29,28,27,26,25,24,23,22,6,5,4,3,2,1,0,14,13,12,11,10,9,8,7],[22,21,20,19,18,17,16,15,29,28,27,26,25,24,23,7,6,5,4,3,2,1,0,14,13,12,11,10,9,8],[23,22,21,20,19,18,17,16,15,29,28,27,26,25,24,8,7,6,5,4,3,2,1,0,14,13,12,11,10,9],[24,23,22,21,20,19,18,17,16,15,29,28,27,26,25,9,8,7,6,5,4,3,2,1,0,14,13,12,11,10],[25,24,23,22,21,20,19,18,17,16,15,29,28,27,26,10,9,8,7,6,5,4,3,2,1,0,14,13,12,11],[26,25,24,23,22,21,20,19,18,17,16,15,29,28,27,11,10,9,8,7,6,5,4,3,2,1,0,14,13,12],[27,26,25,24,23,22,21,20,19,18,17,16,15,29,28,12,11,10,9,8,7,6,5,4,3,2,1,0,14,13],[28,27,26,25,24,23,22,21,20,19,18,17,16,15,29,13,12,11,10,9,8,7,6,5,4,3,2,1,0,14],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0089","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 否, [16, 18, 20, 22, 24, 26, 28, 30], 否, 是, [0, 8], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(16, 否, [16, 18, 20, 22, 24, 26, 28, 30], 否, 是, [0, 8], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152089,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":16,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":16,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,18,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,19,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,20,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,21,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,22,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,23,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,24,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,25,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,26,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,27,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,28,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,29,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,30,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,31,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,18,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,19,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4,3],[19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,20,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5,4],[20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,21,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6,5],[21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,22,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7,6],[22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,23,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8,7],[23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,24,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9,8],[24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,25,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10,9],[25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,26,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11,10],[26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,27,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12,11],[27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,28,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13,12],[28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,29,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14,13],[29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,30,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15,14],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,31,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,15],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0090","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 否, [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(17, 否, [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152090,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":17,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":17,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,20,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,21,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,22,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,23,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,24,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,25,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,26,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,27,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,28,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,29,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,30,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,31,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,32,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,33,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,20,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,21,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5,4],[21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,22,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6,5],[22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,23,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7,6],[23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,24,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8,7],[24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,25,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9,8],[25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,26,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10,9],[26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,27,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11,10],[27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,28,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12,11],[28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,29,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13,12],[29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,30,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14,13],[30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,31,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15,14],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,32,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16,15],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,33,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,16],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0091","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 是, [9], 是, 是, [0, 9], 否, 否, 是, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(9, 是, [9], 是, 是, [0, 9], 否, 否, 是, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152091,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":9,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":9,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,22,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,23,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,24,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,25,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,26,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,27,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,28,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,29,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,30,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,31,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,33,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,34,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,35,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,22,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,23,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5],[23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,24,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6],[24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,25,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7],[25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,26,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8],[26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,27,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9],[27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,28,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11,10],[28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,29,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12,11],[29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,30,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13,12],[30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,31,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14,13],[31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,32,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15,14],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,33,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16,15],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,34,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17,16],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,35,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,17],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0092","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 否, [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(19, 否, [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152092,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":19,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":19,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,24,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,25,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,26,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,27,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,28,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,29,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,30,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,31,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,32,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,33,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,34,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,35,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,36,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,37,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,24,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,25,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7,6],[25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,26,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8,7],[26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,27,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9,8],[27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,28,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10,9],[28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,29,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11,10],[29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,30,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12,11],[30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,31,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13,12],[31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,32,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14,13],[32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,33,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15,14],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,34,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16,15],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,35,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17,16],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,36,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18,17],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,37,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,18],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0093","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 否, [20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 否, 是, [0, 10], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(20, 否, [20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 否, 是, [0, 10], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152093,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":20,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":20,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,26,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,27,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,28,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,29,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,30,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,31,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,32,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,33,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,34,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,35,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,36,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,37,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,38,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,39,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,26,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,27,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8,7],[27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,28,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9,8],[28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,29,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10,9],[29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,30,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11,10],[30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,31,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12,11],[31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,32,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13,12],[32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,33,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14,13],[33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,34,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15,14],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,35,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16,15],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,36,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17,16],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,37,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18,17],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,38,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19,18],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,39,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,19],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0094","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 否, [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(21, 否, [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152094,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":21,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":21,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,28,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,29,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,30,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,31,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,32,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,33,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,34,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,35,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,36,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,37,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,38,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,39,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,40,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,41,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,28,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,29,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9,8],[29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,30,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10,9],[30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,31,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11,10],[31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,32,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12,11],[32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,33,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13,12],[33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,34,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14,13],[34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,35,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15,14],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,36,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16,15],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,37,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17,16],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,38,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18,17],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,39,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19,18],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,40,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20,19],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,41,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,20],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":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、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0095","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12],[34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 否, [22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 否, 是, [0, 11], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(22, 否, [22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 否, 是, [0, 11], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152095,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12],[34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13],[35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,41,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20,19],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,42,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21,20],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,43,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,21],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":22,"domain":"Complete_finite_group_center","operation":"center-abelian-conjugacy-boundary","semantic_template":"qf-mod46-center-abelian-conjugacy-boundary/{\"element\":22,\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,30,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,31,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,32,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,33,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,34,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,35,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,36,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,37,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,38,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,39,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,40,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,41,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,42,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,43,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,30,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,31,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10,9],[31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,32,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11,10],[32,31,30,29,28,27,26,25,24,23,22,43,42,41,40,39,38,37,36,35,34,33,10,9,8,7,6,5,4,3,2,1,0,21,20,19,18,17,16,15,14,13,12,11],[33,32,31,30,29,28,27,26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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0096","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10],[33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11],[34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12],[35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 否, [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(23, 否, [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152096,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,32,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,33,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,34,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,35,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,36,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,37,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,38,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,39,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,40,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,41,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,42,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,43,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,44,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,32,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,33,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11,10],[33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,34,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12,11],[34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,35,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13,12],[35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,36,12,11,10,9,8,7,6,5,4,3,2,1,0,22,21,20,19,18,17,16,15,14,13],[36,35,34,33,32,31,30,29,28,27,26,25,24,23,45,44,43,42,41,40,39,38,37,13,12,11,10,9,8,7,6,5,4,3,2,1,0,22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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0097","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11],[35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12],[36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13],[37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,45,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22,21],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,46,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23,22],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,47,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,23],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 是, [12], 是, 是, [0, 12], 否, 否, 是, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(12, 是, [12], 是, 是, [0, 12], 否, 否, 是, null, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152097,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,35,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,36,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,38,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,39,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,41,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,42,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,43,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,44,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,45,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,46,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,47,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,34,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[34,33,32,31,30,29,28,27,26,25,24,47,46,45,44,43,42,41,40,39,38,37,36,35,10,9,8,7,6,5,4,3,2,1,0,23,22,21,20,19,18,17,16,15,14,13,12,11],[35,34,33,32,31,30,29,28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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0098","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12],[37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13],[38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20,19],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,45,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21,20],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,46,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22,21],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,47,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23,22],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,48,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24,23],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,49,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,24],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 否, [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], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(25, 否, [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], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152098,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,36,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,37,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,38,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,39,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,40,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,41,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,42,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,43,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,44,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,45,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,46,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,47,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,48,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,49,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,7,6,5,4,3,2,1,0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[33,32,31,30,29,28,27,26,25,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,8,7,6,5,4,3,2,1,0,24,2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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0099","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13],[39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19,18],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,45,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20,19],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,46,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21,20],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,47,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22,21],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,48,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23,22],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,49,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24,23],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,50,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25,24],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,51,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,25],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(26, 否, [26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 否, 是, [0, 13], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(26, 否, [26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 否, 是, [0, 13], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152099,"parameters":{"group_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],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,38,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,39,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,40,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,41,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,42,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,43,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,44,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,45,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,46,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,47,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,48,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,49,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,50,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,51,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[31,30,29,28,27,26,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,5,4,3,2,1,0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[32,31,30,29,28,27,26,51,50,49,48,47,46,45,44,43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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r02-finite-group-center-0100","batch_id":"batch-20261015-advance-20261115-r02-finite-group-center-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-group-center"],"title_zh":"有限群中心","stem_zh":"完整Cayley table上的forall-g commutation與center資格。操作=center-abelian-conjugacy-boundary；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5],[32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6],[33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7],[34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8],[35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9],[36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10],[37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11],[38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12],[39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13],[40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14],[41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15],[42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16],[43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,44,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18,17],[44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,45,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19,18],[45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,46,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20,19],[46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,47,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21,20],[47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,48,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22,21],[48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,49,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23,22],[49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,50,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24,23],[50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,51,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25,24],[51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,52,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26,25],[52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,53,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,26],[53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],\"identity\":0,\"element\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 否, [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, 52, 53], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)","explanation_zh":"Wave6第九層核對=(27, 否, [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, 52, 53], 否, 是, [0], 否, 否, 是, 1, center_is_not_a_single_element_centralizer_and_no_class_equation_or_characteristic_claim)；依完整明示資料獨立重算；identity/product/inverse closure與conjugation normality均完整枚舉；central恰當conjugacy class singleton；Z(G)=G恰當G abelian；不混同centralizer且不宣稱class equation或characteristic。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限群中心"]},"variant":{"family_id":"qf-mod46-center-abelian-conjugacy-boundary","seed":202611152100,"parameters":{"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,40,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,41,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,42,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,43,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,44,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,45,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,47,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,48,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,49,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,50,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,51,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,52,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,53,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52],[27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],[28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2],[29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3],[30,29,28,27,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,3,2,1,0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4],[31,30,29,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duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}