{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0001","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"271b3b06cf061ed9cacbecc26508d33d7cc2e2e0b28c3f3269164005703b9b9a","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141001,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,2,1],\"source_identity\":0,\"source_table\":[[0,1,2],[1,2,0],[2,0,1]],\"target_identity\":0,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], source identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], target identity=0, and total mapping=[0, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(3, 3, [0, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 1, 1, Yes], [1, 0, 2, 2, Yes], [1, 1, 1, 1, Yes], [1, 2, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 0, 0, Yes], [2, 2, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (3, 3, [0, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 1, 1, Yes], [1, 0, 2, 2, Yes], [1, 1, 1, 1, Yes], [1, 2, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 0, 0, Yes], [2, 2, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0002","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3473c69c9d4c58e65487956c23036e6b098ac71ce3d8a0964207d630909fe487","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141002,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"target_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], source identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], target identity=0, and total mapping=[0, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(4, 4, [0, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 2, 2, Yes], [0, 3, 1, 1, Yes], [1, 0, 3, 3, Yes], [1, 1, 2, 2, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (4, 4, [0, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 2, 2, Yes], [0, 3, 1, 1, Yes], [1, 0, 3, 3, Yes], [1, 1, 2, 2, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0003","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88ccbbcec88039bbe4f1e6455e64ae84059cabacb8770778d3be6aa3593a8f3a","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141003,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], source identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], target identity=0, and total mapping=[0, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(5, 5, [0, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 3, 3, Yes], [0, 3, 2, 2, Yes], [0, 4, 1, 1, Yes], [1, 0, 4, 4, Yes], [1, 1, 3, 3, Yes], [1, 2, 2, 2, Yes], [1, 3, 1, 1, Yes], [1, 4, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 1, 1, Yes], [2, 3, 0, 0, Yes], [2, 4, 4, 4, Yes], [3, 0, 2, 2, Yes], [3, 1, 1, 1, Yes], [3, 2, 0, 0, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [4, 0, 1, 1, Yes], [4, 1, 0, 0, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (5, 5, [0, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 3, 3, Yes], [0, 3, 2, 2, Yes], [0, 4, 1, 1, Yes], [1, 0, 4, 4, Yes], [1, 1, 3, 3, Yes], [1, 2, 2, 2, Yes], [1, 3, 1, 1, Yes], [1, 4, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 1, 1, Yes], [2, 3, 0, 0, Yes], [2, 4, 4, 4, Yes], [3, 0, 2, 2, Yes], [3, 1, 1, 1, Yes], [3, 2, 0, 0, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [4, 0, 1, 1, Yes], [4, 1, 0, 0, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0004","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d579ee6e413ed597885bdb28b41844e9e53333dfd3ad2a17505c91c561ee98e2","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141004,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], source identity=0, target table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], target identity=0, and total mapping=[0, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(6, 6, [0, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 4, 4, Yes], [0, 3, 3, 3, Yes], [0, 4, 2, 2, Yes], [0, 5, 1, 1, Yes], [1, 0, 5, 5, Yes], [1, 1, 4, 4, Yes], [1, 2, 3, 3, Yes], [1, 3, 2, 2, Yes], [1, 4, 1, 1, Yes], [1, 5, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 3, 3, Yes], [2, 2, 2, 2, Yes], [2, 3, 1, 1, Yes], [2, 4, 0, 0, Yes], [2, 5, 5, 5, Yes], [3, 0, 3, 3, Yes], [3, 1, 2, 2, Yes], [3, 2, 1, 1, Yes], [3, 3, 0, 0, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [4, 0, 2, 2, Yes], [4, 1, 1, 1, Yes], [4, 2, 0, 0, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [5, 0, 1, 1, Yes], [5, 1, 0, 0, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (6, 6, [0, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 4, 4, Yes], [0, 3, 3, 3, Yes], [0, 4, 2, 2, Yes], [0, 5, 1, 1, Yes], [1, 0, 5, 5, Yes], [1, 1, 4, 4, Yes], [1, 2, 3, 3, Yes], [1, 3, 2, 2, Yes], [1, 4, 1, 1, Yes], [1, 5, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 3, 3, Yes], [2, 2, 2, 2, Yes], [2, 3, 1, 1, Yes], [2, 4, 0, 0, Yes], [2, 5, 5, 5, Yes], [3, 0, 3, 3, Yes], [3, 1, 2, 2, Yes], [3, 2, 1, 1, Yes], [3, 3, 0, 0, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [4, 0, 2, 2, Yes], [4, 1, 1, 1, Yes], [4, 2, 0, 0, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [5, 0, 1, 1, Yes], [5, 1, 0, 0, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0005","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23b0063ba800afe56d2c89cfa035d6649db7519c58263b603ef7fae02aecef11","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141005,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], target identity=0, and total mapping=[0, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(7, 7, [0, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 6, 6, Yes], [0, 2, 5, 5, Yes], [0, 3, 4, 4, Yes], [0, 4, 3, 3, Yes], [0, 5, 2, 2, Yes], [0, 6, 1, 1, Yes], [1, 0, 6, 6, Yes], [1, 1, 5, 5, Yes], [1, 2, 4, 4, Yes], [1, 3, 3, 3, Yes], [1, 4, 2, 2, Yes], [1, 5, 1, 1, Yes], [1, 6, 0, 0, Yes], [2, 0, 5, 5, Yes], [2, 1, 4, 4, Yes], [2, 2, 3, 3, Yes], [2, 3, 2, 2, Yes], [2, 4, 1, 1, Yes], [2, 5, 0, 0, Yes], [2, 6, 6, 6, Yes], [3, 0, 4, 4, Yes], [3, 1, 3, 3, Yes], [3, 2, 2, 2, Yes], [3, 3, 1, 1, Yes], [3, 4, 0, 0, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [4, 0, 3, 3, Yes], [4, 1, 2, 2, Yes], [4, 2, 1, 1, Yes], [4, 3, 0, 0, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [5, 0, 2, 2, Yes], [5, 1, 1, 1, Yes], [5, 2, 0, 0, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [6, 0, 1, 1, Yes], [6, 1, 0, 0, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (7, 7, [0, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 6, 6, Yes], [0, 2, 5, 5, Yes], [0, 3, 4, 4, Yes], [0, 4, 3, 3, Yes], [0, 5, 2, 2, Yes], [0, 6, 1, 1, Yes], [1, 0, 6, 6, Yes], [1, 1, 5, 5, Yes], [1, 2, 4, 4, Yes], [1, 3, 3, 3, Yes], [1, 4, 2, 2, Yes], [1, 5, 1, 1, Yes], [1, 6, 0, 0, Yes], [2, 0, 5, 5, Yes], [2, 1, 4, 4, Yes], [2, 2, 3, 3, Yes], [2, 3, 2, 2, Yes], [2, 4, 1, 1, Yes], [2, 5, 0, 0, Yes], [2, 6, 6, 6, Yes], [3, 0, 4, 4, Yes], [3, 1, 3, 3, Yes], [3, 2, 2, 2, Yes], [3, 3, 1, 1, Yes], [3, 4, 0, 0, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [4, 0, 3, 3, Yes], [4, 1, 2, 2, Yes], [4, 2, 1, 1, Yes], [4, 3, 0, 0, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [5, 0, 2, 2, Yes], [5, 1, 1, 1, Yes], [5, 2, 0, 0, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [6, 0, 1, 1, Yes], [6, 1, 0, 0, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0006","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"398934357d5d373850c3d665eec821a9788bf207733b393e4a31ce09a29fd03c","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141006,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], target identity=0, and total mapping=[0, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(8, 8, [0, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 7, 7, Yes], [0, 2, 6, 6, Yes], [0, 3, 5, 5, Yes], [0, 4, 4, 4, Yes], [0, 5, 3, 3, Yes], [0, 6, 2, 2, Yes], [0, 7, 1, 1, Yes], [1, 0, 7, 7, Yes], [1, 1, 6, 6, Yes], [1, 2, 5, 5, Yes], [1, 3, 4, 4, Yes], [1, 4, 3, 3, Yes], [1, 5, 2, 2, Yes], [1, 6, 1, 1, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 5, 5, Yes], [2, 2, 4, 4, Yes], [2, 3, 3, 3, Yes], [2, 4, 2, 2, Yes], [2, 5, 1, 1, Yes], [2, 6, 0, 0, Yes], [2, 7, 7, 7, Yes], [3, 0, 5, 5, Yes], [3, 1, 4, 4, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes], [3, 4, 1, 1, Yes], [3, 5, 0, 0, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 3, 3, Yes], [4, 2, 2, 2, Yes], [4, 3, 1, 1, Yes], [4, 4, 0, 0, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [5, 0, 3, 3, Yes], [5, 1, 2, 2, Yes], [5, 2, 1, 1, Yes], [5, 3, 0, 0, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 1, 1, Yes], [6, 2, 0, 0, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [7, 0, 1, 1, Yes], [7, 1, 0, 0, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (8, 8, [0, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 7, 7, Yes], [0, 2, 6, 6, Yes], [0, 3, 5, 5, Yes], [0, 4, 4, 4, Yes], [0, 5, 3, 3, Yes], [0, 6, 2, 2, Yes], [0, 7, 1, 1, Yes], [1, 0, 7, 7, Yes], [1, 1, 6, 6, Yes], [1, 2, 5, 5, Yes], [1, 3, 4, 4, Yes], [1, 4, 3, 3, Yes], [1, 5, 2, 2, Yes], [1, 6, 1, 1, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 5, 5, Yes], [2, 2, 4, 4, Yes], [2, 3, 3, 3, Yes], [2, 4, 2, 2, Yes], [2, 5, 1, 1, Yes], [2, 6, 0, 0, Yes], [2, 7, 7, 7, Yes], [3, 0, 5, 5, Yes], [3, 1, 4, 4, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes], [3, 4, 1, 1, Yes], [3, 5, 0, 0, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 3, 3, Yes], [4, 2, 2, 2, Yes], [4, 3, 1, 1, Yes], [4, 4, 0, 0, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [5, 0, 3, 3, Yes], [5, 1, 2, 2, Yes], [5, 2, 1, 1, Yes], [5, 3, 0, 0, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 1, 1, Yes], [6, 2, 0, 0, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [7, 0, 1, 1, Yes], [7, 1, 0, 0, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0007","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cd657193b553648383c54eacd4fb13d420bb3b7a2c9f11d9087b03f8feb368a","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141007,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], target identity=0, and total mapping=[0, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(9, 9, [0, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 8, 8, Yes], [0, 2, 7, 7, Yes], [0, 3, 6, 6, Yes], [0, 4, 5, 5, Yes], [0, 5, 4, 4, Yes], [0, 6, 3, 3, Yes], [0, 7, 2, 2, Yes], [0, 8, 1, 1, Yes], [1, 0, 8, 8, Yes], [1, 1, 7, 7, Yes], [1, 2, 6, 6, Yes], [1, 3, 5, 5, Yes], [1, 4, 4, 4, Yes], [1, 5, 3, 3, Yes], [1, 6, 2, 2, Yes], [1, 7, 1, 1, Yes], [1, 8, 0, 0, Yes], [2, 0, 7, 7, Yes], [2, 1, 6, 6, Yes], [2, 2, 5, 5, Yes], [2, 3, 4, 4, Yes], [2, 4, 3, 3, Yes], [2, 5, 2, 2, Yes], [2, 6, 1, 1, Yes], [2, 7, 0, 0, Yes], [2, 8, 8, 8, Yes], [3, 0, 6, 6, Yes], [3, 1, 5, 5, Yes], [3, 2, 4, 4, Yes], [3, 3, 3, 3, Yes], [3, 4, 2, 2, Yes], [3, 5, 1, 1, Yes], [3, 6, 0, 0, Yes], [3, 7, 8, 8, Yes], [3, 8, 7, 7, Yes], [4, 0, 5, 5, Yes], [4, 1, 4, 4, Yes], [4, 2, 3, 3, Yes], [4, 3, 2, 2, Yes], [4, 4, 1, 1, Yes], [4, 5, 0, 0, Yes], [4, 6, 8, 8, Yes], [4, 7, 7, 7, Yes], [4, 8, 6, 6, Yes], [5, 0, 4, 4, Yes], [5, 1, 3, 3, Yes], [5, 2, 2, 2, Yes], [5, 3, 1, 1, Yes], [5, 4, 0, 0, Yes], [5, 5, 8, 8, Yes], [5, 6, 7, 7, Yes], [5, 7, 6, 6, Yes], [5, 8, 5, 5, Yes], [6, 0, 3, 3, Yes], [6, 1, 2, 2, Yes], [6, 2, 1, 1, Yes], [6, 3, 0, 0, Yes], [6, 4, 8, 8, Yes], [6, 5, 7, 7, Yes], [6, 6, 6, 6, Yes], [6, 7, 5, 5, Yes], [6, 8, 4, 4, Yes], [7, 0, 2, 2, Yes], [7, 1, 1, 1, Yes], [7, 2, 0, 0, Yes], [7, 3, 8, 8, Yes], [7, 4, 7, 7, Yes], [7, 5, 6, 6, Yes], [7, 6, 5, 5, Yes], [7, 7, 4, 4, Yes], [7, 8, 3, 3, Yes], [8, 0, 1, 1, Yes], [8, 1, 0, 0, Yes], [8, 2, 8, 8, Yes], [8, 3, 7, 7, Yes], [8, 4, 6, 6, Yes], [8, 5, 5, 5, Yes], [8, 6, 4, 4, Yes], [8, 7, 3, 3, Yes], [8, 8, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (9, 9, [0, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 8, 8, Yes], [0, 2, 7, 7, Yes], [0, 3, 6, 6, Yes], [0, 4, 5, 5, Yes], [0, 5, 4, 4, Yes], [0, 6, 3, 3, Yes], [0, 7, 2, 2, Yes], [0, 8, 1, 1, Yes], [1, 0, 8, 8, Yes], [1, 1, 7, 7, Yes], [1, 2, 6, 6, Yes], [1, 3, 5, 5, Yes], [1, 4, 4, 4, Yes], [1, 5, 3, 3, Yes], [1, 6, 2, 2, Yes], [1, 7, 1, 1, Yes], [1, 8, 0, 0, Yes], [2, 0, 7, 7, Yes], [2, 1, 6, 6, Yes], [2, 2, 5, 5, Yes], [2, 3, 4, 4, Yes], [2, 4, 3, 3, Yes], [2, 5, 2, 2, Yes], [2, 6, 1, 1, Yes], [2, 7, 0, 0, Yes], [2, 8, 8, 8, Yes], [3, 0, 6, 6, Yes], [3, 1, 5, 5, Yes], [3, 2, 4, 4, Yes], [3, 3, 3, 3, Yes], [3, 4, 2, 2, Yes], [3, 5, 1, 1, Yes], [3, 6, 0, 0, Yes], [3, 7, 8, 8, Yes], [3, 8, 7, 7, Yes], [4, 0, 5, 5, Yes], [4, 1, 4, 4, Yes], [4, 2, 3, 3, Yes], [4, 3, 2, 2, Yes], [4, 4, 1, 1, Yes], [4, 5, 0, 0, Yes], [4, 6, 8, 8, Yes], [4, 7, 7, 7, Yes], [4, 8, 6, 6, Yes], [5, 0, 4, 4, Yes], [5, 1, 3, 3, Yes], [5, 2, 2, 2, Yes], [5, 3, 1, 1, Yes], [5, 4, 0, 0, Yes], [5, 5, 8, 8, Yes], [5, 6, 7, 7, Yes], [5, 7, 6, 6, Yes], [5, 8, 5, 5, Yes], [6, 0, 3, 3, Yes], [6, 1, 2, 2, Yes], [6, 2, 1, 1, Yes], [6, 3, 0, 0, Yes], [6, 4, 8, 8, Yes], [6, 5, 7, 7, Yes], [6, 6, 6, 6, Yes], [6, 7, 5, 5, Yes], [6, 8, 4, 4, Yes], [7, 0, 2, 2, Yes], [7, 1, 1, 1, Yes], [7, 2, 0, 0, Yes], [7, 3, 8, 8, Yes], [7, 4, 7, 7, Yes], [7, 5, 6, 6, Yes], [7, 6, 5, 5, Yes], [7, 7, 4, 4, Yes], [7, 8, 3, 3, Yes], [8, 0, 1, 1, Yes], [8, 1, 0, 0, Yes], [8, 2, 8, 8, Yes], [8, 3, 7, 7, Yes], [8, 4, 6, 6, Yes], [8, 5, 5, 5, Yes], [8, 6, 4, 4, Yes], [8, 7, 3, 3, Yes], [8, 8, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0008","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"629f53adb33a060ebadfcc4f467f2a09a550237e4fd2b90df007f416886617a9","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141008,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], target identity=0, and total mapping=[0, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(10, 10, [0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 9, 9, Yes], [0, 2, 8, 8, Yes], [0, 3, 7, 7, Yes], [0, 4, 6, 6, Yes], [0, 5, 5, 5, Yes], [0, 6, 4, 4, Yes], [0, 7, 3, 3, Yes], [0, 8, 2, 2, Yes], [0, 9, 1, 1, Yes], [1, 0, 9, 9, Yes], [1, 1, 8, 8, Yes], [1, 2, 7, 7, Yes], [1, 3, 6, 6, Yes], [1, 4, 5, 5, Yes], [1, 5, 4, 4, Yes], [1, 6, 3, 3, Yes], [1, 7, 2, 2, Yes], [1, 8, 1, 1, Yes], [1, 9, 0, 0, Yes], [2, 0, 8, 8, Yes], [2, 1, 7, 7, Yes], [2, 2, 6, 6, Yes], [2, 3, 5, 5, Yes], [2, 4, 4, 4, Yes], [2, 5, 3, 3, Yes], [2, 6, 2, 2, Yes], [2, 7, 1, 1, Yes], [2, 8, 0, 0, Yes], [2, 9, 9, 9, Yes], [3, 0, 7, 7, Yes], [3, 1, 6, 6, Yes], [3, 2, 5, 5, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [3, 5, 2, 2, Yes], [3, 6, 1, 1, Yes], [3, 7, 0, 0, Yes], [3, 8, 9, 9, Yes], [3, 9, 8, 8, Yes], [4, 0, 6, 6, Yes], [4, 1, 5, 5, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes], [4, 5, 1, 1, Yes], [4, 6, 0, 0, Yes], [4, 7, 9, 9, Yes], [4, 8, 8, 8, Yes], [4, 9, 7, 7, Yes], [5, 0, 5, 5, Yes], [5, 1, 4, 4, Yes], [5, 2, 3, 3, Yes], [5, 3, 2, 2, Yes], [5, 4, 1, 1, Yes], [5, 5, 0, 0, Yes], [5, 6, 9, 9, Yes], [5, 7, 8, 8, Yes], [5, 8, 7, 7, Yes], [5, 9, 6, 6, Yes], [6, 0, 4, 4, Yes], [6, 1, 3, 3, Yes], [6, 2, 2, 2, Yes], [6, 3, 1, 1, Yes], [6, 4, 0, 0, Yes], [6, 5, 9, 9, Yes], [6, 6, 8, 8, Yes], [6, 7, 7, 7, Yes], [6, 8, 6, 6, Yes], [6, 9, 5, 5, Yes], [7, 0, 3, 3, Yes], [7, 1, 2, 2, Yes], [7, 2, 1, 1, Yes], [7, 3, 0, 0, Yes], [7, 4, 9, 9, Yes], [7, 5, 8, 8, Yes], [7, 6, 7, 7, Yes], [7, 7, 6, 6, Yes], [7, 8, 5, 5, Yes], [7, 9, 4, 4, Yes], [8, 0, 2, 2, Yes], [8, 1, 1, 1, Yes], [8, 2, 0, 0, Yes], [8, 3, 9, 9, Yes], [8, 4, 8, 8, Yes], [8, 5, 7, 7, Yes], [8, 6, 6, 6, Yes], [8, 7, 5, 5, Yes], [8, 8, 4, 4, Yes], [8, 9, 3, 3, Yes], [9, 0, 1, 1, Yes], [9, 1, 0, 0, Yes], [9, 2, 9, 9, Yes], [9, 3, 8, 8, Yes], [9, 4, 7, 7, Yes], [9, 5, 6, 6, Yes], [9, 6, 5, 5, Yes], [9, 7, 4, 4, Yes], [9, 8, 3, 3, Yes], [9, 9, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (10, 10, [0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 9, 9, Yes], [0, 2, 8, 8, Yes], [0, 3, 7, 7, Yes], [0, 4, 6, 6, Yes], [0, 5, 5, 5, Yes], [0, 6, 4, 4, Yes], [0, 7, 3, 3, Yes], [0, 8, 2, 2, Yes], [0, 9, 1, 1, Yes], [1, 0, 9, 9, Yes], [1, 1, 8, 8, Yes], [1, 2, 7, 7, Yes], [1, 3, 6, 6, Yes], [1, 4, 5, 5, Yes], [1, 5, 4, 4, Yes], [1, 6, 3, 3, Yes], [1, 7, 2, 2, Yes], [1, 8, 1, 1, Yes], [1, 9, 0, 0, Yes], [2, 0, 8, 8, Yes], [2, 1, 7, 7, Yes], [2, 2, 6, 6, Yes], [2, 3, 5, 5, Yes], [2, 4, 4, 4, Yes], [2, 5, 3, 3, Yes], [2, 6, 2, 2, Yes], [2, 7, 1, 1, Yes], [2, 8, 0, 0, Yes], [2, 9, 9, 9, Yes], [3, 0, 7, 7, Yes], [3, 1, 6, 6, Yes], [3, 2, 5, 5, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [3, 5, 2, 2, Yes], [3, 6, 1, 1, Yes], [3, 7, 0, 0, Yes], [3, 8, 9, 9, Yes], [3, 9, 8, 8, Yes], [4, 0, 6, 6, Yes], [4, 1, 5, 5, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes], [4, 5, 1, 1, Yes], [4, 6, 0, 0, Yes], [4, 7, 9, 9, Yes], [4, 8, 8, 8, Yes], [4, 9, 7, 7, Yes], [5, 0, 5, 5, Yes], [5, 1, 4, 4, Yes], [5, 2, 3, 3, Yes], [5, 3, 2, 2, Yes], [5, 4, 1, 1, Yes], [5, 5, 0, 0, Yes], [5, 6, 9, 9, Yes], [5, 7, 8, 8, Yes], [5, 8, 7, 7, Yes], [5, 9, 6, 6, Yes], [6, 0, 4, 4, Yes], [6, 1, 3, 3, Yes], [6, 2, 2, 2, Yes], [6, 3, 1, 1, Yes], [6, 4, 0, 0, Yes], [6, 5, 9, 9, Yes], [6, 6, 8, 8, Yes], [6, 7, 7, 7, Yes], [6, 8, 6, 6, Yes], [6, 9, 5, 5, Yes], [7, 0, 3, 3, Yes], [7, 1, 2, 2, Yes], [7, 2, 1, 1, Yes], [7, 3, 0, 0, Yes], [7, 4, 9, 9, Yes], [7, 5, 8, 8, Yes], [7, 6, 7, 7, Yes], [7, 7, 6, 6, Yes], [7, 8, 5, 5, Yes], [7, 9, 4, 4, Yes], [8, 0, 2, 2, Yes], [8, 1, 1, 1, Yes], [8, 2, 0, 0, Yes], [8, 3, 9, 9, Yes], [8, 4, 8, 8, Yes], [8, 5, 7, 7, Yes], [8, 6, 6, 6, Yes], [8, 7, 5, 5, Yes], [8, 8, 4, 4, Yes], [8, 9, 3, 3, Yes], [9, 0, 1, 1, Yes], [9, 1, 0, 0, Yes], [9, 2, 9, 9, Yes], [9, 3, 8, 8, Yes], [9, 4, 7, 7, Yes], [9, 5, 6, 6, Yes], [9, 6, 5, 5, Yes], [9, 7, 4, 4, Yes], [9, 8, 3, 3, Yes], [9, 9, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0009","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e485b1897988d217c025cb47c361fb48d30f34047181e6a745ab1e47c1c7a714","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141009,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], target identity=0, and total mapping=[0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(11, 11, [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 10, 10, Yes], [0, 2, 9, 9, Yes], [0, 3, 8, 8, Yes], [0, 4, 7, 7, Yes], [0, 5, 6, 6, Yes], [0, 6, 5, 5, Yes], [0, 7, 4, 4, Yes], [0, 8, 3, 3, Yes], [0, 9, 2, 2, Yes], [0, 10, 1, 1, Yes], [1, 0, 10, 10, Yes], [1, 1, 9, 9, Yes], [1, 2, 8, 8, Yes], [1, 3, 7, 7, Yes], [1, 4, 6, 6, Yes], [1, 5, 5, 5, Yes], [1, 6, 4, 4, Yes], [1, 7, 3, 3, Yes], [1, 8, 2, 2, Yes], [1, 9, 1, 1, Yes], [1, 10, 0, 0, Yes], [2, 0, 9, 9, Yes], [2, 1, 8, 8, Yes], [2, 2, 7, 7, Yes], [2, 3, 6, 6, Yes], [2, 4, 5, 5, Yes], [2, 5, 4, 4, Yes], [2, 6, 3, 3, Yes], [2, 7, 2, 2, Yes], [2, 8, 1, 1, Yes], [2, 9, 0, 0, Yes], [2, 10, 10, 10, Yes], [3, 0, 8, 8, Yes], [3, 1, 7, 7, Yes], [3, 2, 6, 6, Yes], [3, 3, 5, 5, Yes], [3, 4, 4, 4, Yes], [3, 5, 3, 3, Yes], [3, 6, 2, 2, Yes], [3, 7, 1, 1, Yes], [3, 8, 0, 0, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [4, 0, 7, 7, Yes], [4, 1, 6, 6, Yes], [4, 2, 5, 5, Yes], [4, 3, 4, 4, Yes], [4, 4, 3, 3, Yes], [4, 5, 2, 2, Yes], [4, 6, 1, 1, Yes], [4, 7, 0, 0, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [5, 0, 6, 6, Yes], [5, 1, 5, 5, Yes], [5, 2, 4, 4, Yes], [5, 3, 3, 3, Yes], [5, 4, 2, 2, Yes], [5, 5, 1, 1, Yes], [5, 6, 0, 0, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [6, 0, 5, 5, Yes], [6, 1, 4, 4, Yes], [6, 2, 3, 3, Yes], [6, 3, 2, 2, Yes], [6, 4, 1, 1, Yes], [6, 5, 0, 0, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [7, 0, 4, 4, Yes], [7, 1, 3, 3, Yes], [7, 2, 2, 2, Yes], [7, 3, 1, 1, Yes], [7, 4, 0, 0, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [8, 0, 3, 3, Yes], [8, 1, 2, 2, Yes], [8, 2, 1, 1, Yes], [8, 3, 0, 0, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [9, 0, 2, 2, Yes], [9, 1, 1, 1, Yes], [9, 2, 0, 0, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [10, 0, 1, 1, Yes], [10, 1, 0, 0, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (11, 11, [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 10, 10, Yes], [0, 2, 9, 9, Yes], [0, 3, 8, 8, Yes], [0, 4, 7, 7, Yes], [0, 5, 6, 6, Yes], [0, 6, 5, 5, Yes], [0, 7, 4, 4, Yes], [0, 8, 3, 3, Yes], [0, 9, 2, 2, Yes], [0, 10, 1, 1, Yes], [1, 0, 10, 10, Yes], [1, 1, 9, 9, Yes], [1, 2, 8, 8, Yes], [1, 3, 7, 7, Yes], [1, 4, 6, 6, Yes], [1, 5, 5, 5, Yes], [1, 6, 4, 4, Yes], [1, 7, 3, 3, Yes], [1, 8, 2, 2, Yes], [1, 9, 1, 1, Yes], [1, 10, 0, 0, Yes], [2, 0, 9, 9, Yes], [2, 1, 8, 8, Yes], [2, 2, 7, 7, Yes], [2, 3, 6, 6, Yes], [2, 4, 5, 5, Yes], [2, 5, 4, 4, Yes], [2, 6, 3, 3, Yes], [2, 7, 2, 2, Yes], [2, 8, 1, 1, Yes], [2, 9, 0, 0, Yes], [2, 10, 10, 10, Yes], [3, 0, 8, 8, Yes], [3, 1, 7, 7, Yes], [3, 2, 6, 6, Yes], [3, 3, 5, 5, Yes], [3, 4, 4, 4, Yes], [3, 5, 3, 3, Yes], [3, 6, 2, 2, Yes], [3, 7, 1, 1, Yes], [3, 8, 0, 0, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [4, 0, 7, 7, Yes], [4, 1, 6, 6, Yes], [4, 2, 5, 5, Yes], [4, 3, 4, 4, Yes], [4, 4, 3, 3, Yes], [4, 5, 2, 2, Yes], [4, 6, 1, 1, Yes], [4, 7, 0, 0, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [5, 0, 6, 6, Yes], [5, 1, 5, 5, Yes], [5, 2, 4, 4, Yes], [5, 3, 3, 3, Yes], [5, 4, 2, 2, Yes], [5, 5, 1, 1, Yes], [5, 6, 0, 0, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [6, 0, 5, 5, Yes], [6, 1, 4, 4, Yes], [6, 2, 3, 3, Yes], [6, 3, 2, 2, Yes], [6, 4, 1, 1, Yes], [6, 5, 0, 0, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [7, 0, 4, 4, Yes], [7, 1, 3, 3, Yes], [7, 2, 2, 2, Yes], [7, 3, 1, 1, Yes], [7, 4, 0, 0, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [8, 0, 3, 3, Yes], [8, 1, 2, 2, Yes], [8, 2, 1, 1, Yes], [8, 3, 0, 0, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [9, 0, 2, 2, Yes], [9, 1, 1, 1, Yes], [9, 2, 0, 0, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [10, 0, 1, 1, Yes], [10, 1, 0, 0, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0010","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8f97f02ed2414e1a0f4774529417f2fdab5e37e4451675c4066b75dbb8ac845","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141010,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], target identity=0, and total mapping=[0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(12, 12, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 11, 11, Yes], [0, 2, 10, 10, Yes], [0, 3, 9, 9, Yes], [0, 4, 8, 8, Yes], [0, 5, 7, 7, Yes], [0, 6, 6, 6, Yes], [0, 7, 5, 5, Yes], [0, 8, 4, 4, Yes], [0, 9, 3, 3, Yes], [0, 10, 2, 2, Yes], [0, 11, 1, 1, Yes], [1, 0, 11, 11, Yes], [1, 1, 10, 10, Yes], [1, 2, 9, 9, Yes], [1, 3, 8, 8, Yes], [1, 4, 7, 7, Yes], [1, 5, 6, 6, Yes], [1, 6, 5, 5, Yes], [1, 7, 4, 4, Yes], [1, 8, 3, 3, Yes], [1, 9, 2, 2, Yes], [1, 10, 1, 1, Yes], [1, 11, 0, 0, Yes], [2, 0, 10, 10, Yes], [2, 1, 9, 9, Yes], [2, 2, 8, 8, Yes], [2, 3, 7, 7, Yes], [2, 4, 6, 6, Yes], [2, 5, 5, 5, Yes], [2, 6, 4, 4, Yes], [2, 7, 3, 3, Yes], [2, 8, 2, 2, Yes], [2, 9, 1, 1, Yes], [2, 10, 0, 0, Yes], [2, 11, 11, 11, Yes], [3, 0, 9, 9, Yes], [3, 1, 8, 8, Yes], [3, 2, 7, 7, Yes], [3, 3, 6, 6, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [3, 6, 3, 3, Yes], [3, 7, 2, 2, Yes], [3, 8, 1, 1, Yes], [3, 9, 0, 0, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [4, 0, 8, 8, Yes], [4, 1, 7, 7, Yes], [4, 2, 6, 6, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [4, 6, 2, 2, Yes], [4, 7, 1, 1, Yes], [4, 8, 0, 0, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [5, 0, 7, 7, Yes], [5, 1, 6, 6, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes], [5, 6, 1, 1, Yes], [5, 7, 0, 0, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [6, 0, 6, 6, Yes], [6, 1, 5, 5, Yes], [6, 2, 4, 4, Yes], [6, 3, 3, 3, Yes], [6, 4, 2, 2, Yes], [6, 5, 1, 1, Yes], [6, 6, 0, 0, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 4, 4, Yes], [7, 2, 3, 3, Yes], [7, 3, 2, 2, Yes], [7, 4, 1, 1, Yes], [7, 5, 0, 0, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [8, 0, 4, 4, Yes], [8, 1, 3, 3, Yes], [8, 2, 2, 2, Yes], [8, 3, 1, 1, Yes], [8, 4, 0, 0, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [9, 0, 3, 3, Yes], [9, 1, 2, 2, Yes], [9, 2, 1, 1, Yes], [9, 3, 0, 0, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [10, 0, 2, 2, Yes], [10, 1, 1, 1, Yes], [10, 2, 0, 0, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [11, 0, 1, 1, Yes], [11, 1, 0, 0, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (12, 12, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 11, 11, Yes], [0, 2, 10, 10, Yes], [0, 3, 9, 9, Yes], [0, 4, 8, 8, Yes], [0, 5, 7, 7, Yes], [0, 6, 6, 6, Yes], [0, 7, 5, 5, Yes], [0, 8, 4, 4, Yes], [0, 9, 3, 3, Yes], [0, 10, 2, 2, Yes], [0, 11, 1, 1, Yes], [1, 0, 11, 11, Yes], [1, 1, 10, 10, Yes], [1, 2, 9, 9, Yes], [1, 3, 8, 8, Yes], [1, 4, 7, 7, Yes], [1, 5, 6, 6, Yes], [1, 6, 5, 5, Yes], [1, 7, 4, 4, Yes], [1, 8, 3, 3, Yes], [1, 9, 2, 2, Yes], [1, 10, 1, 1, Yes], [1, 11, 0, 0, Yes], [2, 0, 10, 10, Yes], [2, 1, 9, 9, Yes], [2, 2, 8, 8, Yes], [2, 3, 7, 7, Yes], [2, 4, 6, 6, Yes], [2, 5, 5, 5, Yes], [2, 6, 4, 4, Yes], [2, 7, 3, 3, Yes], [2, 8, 2, 2, Yes], [2, 9, 1, 1, Yes], [2, 10, 0, 0, Yes], [2, 11, 11, 11, Yes], [3, 0, 9, 9, Yes], [3, 1, 8, 8, Yes], [3, 2, 7, 7, Yes], [3, 3, 6, 6, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [3, 6, 3, 3, Yes], [3, 7, 2, 2, Yes], [3, 8, 1, 1, Yes], [3, 9, 0, 0, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [4, 0, 8, 8, Yes], [4, 1, 7, 7, Yes], [4, 2, 6, 6, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [4, 6, 2, 2, Yes], [4, 7, 1, 1, Yes], [4, 8, 0, 0, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [5, 0, 7, 7, Yes], [5, 1, 6, 6, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes], [5, 6, 1, 1, Yes], [5, 7, 0, 0, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [6, 0, 6, 6, Yes], [6, 1, 5, 5, Yes], [6, 2, 4, 4, Yes], [6, 3, 3, 3, Yes], [6, 4, 2, 2, Yes], [6, 5, 1, 1, Yes], [6, 6, 0, 0, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 4, 4, Yes], [7, 2, 3, 3, Yes], [7, 3, 2, 2, Yes], [7, 4, 1, 1, Yes], [7, 5, 0, 0, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [8, 0, 4, 4, Yes], [8, 1, 3, 3, Yes], [8, 2, 2, 2, Yes], [8, 3, 1, 1, Yes], [8, 4, 0, 0, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [9, 0, 3, 3, Yes], [9, 1, 2, 2, Yes], [9, 2, 1, 1, Yes], [9, 3, 0, 0, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [10, 0, 2, 2, Yes], [10, 1, 1, 1, Yes], [10, 2, 0, 0, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [11, 0, 1, 1, Yes], [11, 1, 0, 0, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0011","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb7529099facf15266f734223fe9541d123cd44a8b104e3c7f2d7ee60e9e8673","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141011,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], target identity=0, and total mapping=[0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(13, 13, [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 12, 12, Yes], [0, 2, 11, 11, Yes], [0, 3, 10, 10, Yes], [0, 4, 9, 9, Yes], [0, 5, 8, 8, Yes], [0, 6, 7, 7, Yes], [0, 7, 6, 6, Yes], [0, 8, 5, 5, Yes], [0, 9, 4, 4, Yes], [0, 10, 3, 3, Yes], [0, 11, 2, 2, Yes], [0, 12, 1, 1, Yes], [1, 0, 12, 12, Yes], [1, 1, 11, 11, Yes], [1, 2, 10, 10, Yes], [1, 3, 9, 9, Yes], [1, 4, 8, 8, Yes], [1, 5, 7, 7, Yes], [1, 6, 6, 6, Yes], [1, 7, 5, 5, Yes], [1, 8, 4, 4, Yes], [1, 9, 3, 3, Yes], [1, 10, 2, 2, Yes], [1, 11, 1, 1, Yes], [1, 12, 0, 0, Yes], [2, 0, 11, 11, Yes], [2, 1, 10, 10, Yes], [2, 2, 9, 9, Yes], [2, 3, 8, 8, Yes], [2, 4, 7, 7, Yes], [2, 5, 6, 6, Yes], [2, 6, 5, 5, Yes], [2, 7, 4, 4, Yes], [2, 8, 3, 3, Yes], [2, 9, 2, 2, Yes], [2, 10, 1, 1, Yes], [2, 11, 0, 0, Yes], [2, 12, 12, 12, Yes], [3, 0, 10, 10, Yes], [3, 1, 9, 9, Yes], [3, 2, 8, 8, Yes], [3, 3, 7, 7, Yes], [3, 4, 6, 6, Yes], [3, 5, 5, 5, Yes], [3, 6, 4, 4, Yes], [3, 7, 3, 3, Yes], [3, 8, 2, 2, Yes], [3, 9, 1, 1, Yes], [3, 10, 0, 0, Yes], [3, 11, 12, 12, Yes], [3, 12, 11, 11, Yes], [4, 0, 9, 9, Yes], [4, 1, 8, 8, Yes], [4, 2, 7, 7, Yes], [4, 3, 6, 6, Yes], [4, 4, 5, 5, Yes], [4, 5, 4, 4, Yes], [4, 6, 3, 3, Yes], [4, 7, 2, 2, Yes], [4, 8, 1, 1, Yes], [4, 9, 0, 0, Yes], [4, 10, 12, 12, Yes], [4, 11, 11, 11, Yes], [4, 12, 10, 10, Yes], [5, 0, 8, 8, Yes], [5, 1, 7, 7, Yes], [5, 2, 6, 6, Yes], [5, 3, 5, 5, Yes], [5, 4, 4, 4, Yes], [5, 5, 3, 3, Yes], [5, 6, 2, 2, Yes], [5, 7, 1, 1, Yes], [5, 8, 0, 0, Yes], [5, 9, 12, 12, Yes], [5, 10, 11, 11, Yes], [5, 11, 10, 10, Yes], [5, 12, 9, 9, Yes], [6, 0, 7, 7, Yes], [6, 1, 6, 6, Yes], [6, 2, 5, 5, Yes], [6, 3, 4, 4, Yes], [6, 4, 3, 3, Yes], [6, 5, 2, 2, Yes], [6, 6, 1, 1, Yes], [6, 7, 0, 0, Yes], [6, 8, 12, 12, Yes], [6, 9, 11, 11, Yes], [6, 10, 10, 10, Yes], [6, 11, 9, 9, Yes], [6, 12, 8, 8, Yes], [7, 0, 6, 6, Yes], [7, 1, 5, 5, Yes], [7, 2, 4, 4, Yes], [7, 3, 3, 3, Yes], [7, 4, 2, 2, Yes], [7, 5, 1, 1, Yes], [7, 6, 0, 0, Yes], [7, 7, 12, 12, Yes], [7, 8, 11, 11, Yes], [7, 9, 10, 10, Yes], [7, 10, 9, 9, Yes], [7, 11, 8, 8, Yes], [7, 12, 7, 7, Yes], [8, 0, 5, 5, Yes], [8, 1, 4, 4, Yes], [8, 2, 3, 3, Yes], [8, 3, 2, 2, Yes], [8, 4, 1, 1, Yes], [8, 5, 0, 0, Yes], [8, 6, 12, 12, Yes], [8, 7, 11, 11, Yes], [8, 8, 10, 10, Yes], [8, 9, 9, 9, Yes], [8, 10, 8, 8, Yes], [8, 11, 7, 7, Yes], [8, 12, 6, 6, Yes], [9, 0, 4, 4, Yes], [9, 1, 3, 3, Yes], [9, 2, 2, 2, Yes], [9, 3, 1, 1, Yes], [9, 4, 0, 0, Yes], [9, 5, 12, 12, Yes], [9, 6, 11, 11, Yes], [9, 7, 10, 10, Yes], [9, 8, 9, 9, Yes], [9, 9, 8, 8, Yes], [9, 10, 7, 7, Yes], [9, 11, 6, 6, Yes], [9, 12, 5, 5, Yes], [10, 0, 3, 3, Yes], [10, 1, 2, 2, Yes], [10, 2, 1, 1, Yes], [10, 3, 0, 0, Yes], [10, 4, 12, 12, Yes], [10, 5, 11, 11, Yes], [10, 6, 10, 10, Yes], [10, 7, 9, 9, Yes], [10, 8, 8, 8, Yes], [10, 9, 7, 7, Yes], [10, 10, 6, 6, Yes], [10, 11, 5, 5, Yes], [10, 12, 4, 4, Yes], [11, 0, 2, 2, Yes], [11, 1, 1, 1, Yes], [11, 2, 0, 0, Yes], [11, 3, 12, 12, Yes], [11, 4, 11, 11, Yes], [11, 5, 10, 10, Yes], [11, 6, 9, 9, Yes], [11, 7, 8, 8, Yes], [11, 8, 7, 7, Yes], [11, 9, 6, 6, Yes], [11, 10, 5, 5, Yes], [11, 11, 4, 4, Yes], [11, 12, 3, 3, Yes], [12, 0, 1, 1, Yes], [12, 1, 0, 0, Yes], [12, 2, 12, 12, Yes], [12, 3, 11, 11, Yes], [12, 4, 10, 10, Yes], [12, 5, 9, 9, Yes], [12, 6, 8, 8, Yes], [12, 7, 7, 7, Yes], [12, 8, 6, 6, Yes], [12, 9, 5, 5, Yes], [12, 10, 4, 4, Yes], [12, 11, 3, 3, Yes], [12, 12, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (13, 13, [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 12, 12, Yes], [0, 2, 11, 11, Yes], [0, 3, 10, 10, Yes], [0, 4, 9, 9, Yes], [0, 5, 8, 8, Yes], [0, 6, 7, 7, Yes], [0, 7, 6, 6, Yes], [0, 8, 5, 5, Yes], [0, 9, 4, 4, Yes], [0, 10, 3, 3, Yes], [0, 11, 2, 2, Yes], [0, 12, 1, 1, Yes], [1, 0, 12, 12, Yes], [1, 1, 11, 11, Yes], [1, 2, 10, 10, Yes], [1, 3, 9, 9, Yes], [1, 4, 8, 8, Yes], [1, 5, 7, 7, Yes], [1, 6, 6, 6, Yes], [1, 7, 5, 5, Yes], [1, 8, 4, 4, Yes], [1, 9, 3, 3, Yes], [1, 10, 2, 2, Yes], [1, 11, 1, 1, Yes], [1, 12, 0, 0, Yes], [2, 0, 11, 11, Yes], [2, 1, 10, 10, Yes], [2, 2, 9, 9, Yes], [2, 3, 8, 8, Yes], [2, 4, 7, 7, Yes], [2, 5, 6, 6, Yes], [2, 6, 5, 5, Yes], [2, 7, 4, 4, Yes], [2, 8, 3, 3, Yes], [2, 9, 2, 2, Yes], [2, 10, 1, 1, Yes], [2, 11, 0, 0, Yes], [2, 12, 12, 12, Yes], [3, 0, 10, 10, Yes], [3, 1, 9, 9, Yes], [3, 2, 8, 8, Yes], [3, 3, 7, 7, Yes], [3, 4, 6, 6, Yes], [3, 5, 5, 5, Yes], [3, 6, 4, 4, Yes], [3, 7, 3, 3, Yes], [3, 8, 2, 2, Yes], [3, 9, 1, 1, Yes], [3, 10, 0, 0, Yes], [3, 11, 12, 12, Yes], [3, 12, 11, 11, Yes], [4, 0, 9, 9, Yes], [4, 1, 8, 8, Yes], [4, 2, 7, 7, Yes], [4, 3, 6, 6, Yes], [4, 4, 5, 5, Yes], [4, 5, 4, 4, Yes], [4, 6, 3, 3, Yes], [4, 7, 2, 2, Yes], [4, 8, 1, 1, Yes], [4, 9, 0, 0, Yes], [4, 10, 12, 12, Yes], [4, 11, 11, 11, Yes], [4, 12, 10, 10, Yes], [5, 0, 8, 8, Yes], [5, 1, 7, 7, Yes], [5, 2, 6, 6, Yes], [5, 3, 5, 5, Yes], [5, 4, 4, 4, Yes], [5, 5, 3, 3, Yes], [5, 6, 2, 2, Yes], [5, 7, 1, 1, Yes], [5, 8, 0, 0, Yes], [5, 9, 12, 12, Yes], [5, 10, 11, 11, Yes], [5, 11, 10, 10, Yes], [5, 12, 9, 9, Yes], [6, 0, 7, 7, Yes], [6, 1, 6, 6, Yes], [6, 2, 5, 5, Yes], [6, 3, 4, 4, Yes], [6, 4, 3, 3, Yes], [6, 5, 2, 2, Yes], [6, 6, 1, 1, Yes], [6, 7, 0, 0, Yes], [6, 8, 12, 12, Yes], [6, 9, 11, 11, Yes], [6, 10, 10, 10, Yes], [6, 11, 9, 9, Yes], [6, 12, 8, 8, Yes], [7, 0, 6, 6, Yes], [7, 1, 5, 5, Yes], [7, 2, 4, 4, Yes], [7, 3, 3, 3, Yes], [7, 4, 2, 2, Yes], [7, 5, 1, 1, Yes], [7, 6, 0, 0, Yes], [7, 7, 12, 12, Yes], [7, 8, 11, 11, Yes], [7, 9, 10, 10, Yes], [7, 10, 9, 9, Yes], [7, 11, 8, 8, Yes], [7, 12, 7, 7, Yes], [8, 0, 5, 5, Yes], [8, 1, 4, 4, Yes], [8, 2, 3, 3, Yes], [8, 3, 2, 2, Yes], [8, 4, 1, 1, Yes], [8, 5, 0, 0, Yes], [8, 6, 12, 12, Yes], [8, 7, 11, 11, Yes], [8, 8, 10, 10, Yes], [8, 9, 9, 9, Yes], [8, 10, 8, 8, Yes], [8, 11, 7, 7, Yes], [8, 12, 6, 6, Yes], [9, 0, 4, 4, Yes], [9, 1, 3, 3, Yes], [9, 2, 2, 2, Yes], [9, 3, 1, 1, Yes], [9, 4, 0, 0, Yes], [9, 5, 12, 12, Yes], [9, 6, 11, 11, Yes], [9, 7, 10, 10, Yes], [9, 8, 9, 9, Yes], [9, 9, 8, 8, Yes], [9, 10, 7, 7, Yes], [9, 11, 6, 6, Yes], [9, 12, 5, 5, Yes], [10, 0, 3, 3, Yes], [10, 1, 2, 2, Yes], [10, 2, 1, 1, Yes], [10, 3, 0, 0, Yes], [10, 4, 12, 12, Yes], [10, 5, 11, 11, Yes], [10, 6, 10, 10, Yes], [10, 7, 9, 9, Yes], [10, 8, 8, 8, Yes], [10, 9, 7, 7, Yes], [10, 10, 6, 6, Yes], [10, 11, 5, 5, Yes], [10, 12, 4, 4, Yes], [11, 0, 2, 2, Yes], [11, 1, 1, 1, Yes], [11, 2, 0, 0, Yes], [11, 3, 12, 12, Yes], [11, 4, 11, 11, Yes], [11, 5, 10, 10, Yes], [11, 6, 9, 9, Yes], [11, 7, 8, 8, Yes], [11, 8, 7, 7, Yes], [11, 9, 6, 6, Yes], [11, 10, 5, 5, Yes], [11, 11, 4, 4, Yes], [11, 12, 3, 3, Yes], [12, 0, 1, 1, Yes], [12, 1, 0, 0, Yes], [12, 2, 12, 12, Yes], [12, 3, 11, 11, Yes], [12, 4, 10, 10, Yes], [12, 5, 9, 9, Yes], [12, 6, 8, 8, Yes], [12, 7, 7, 7, Yes], [12, 8, 6, 6, Yes], [12, 9, 5, 5, Yes], [12, 10, 4, 4, Yes], [12, 11, 3, 3, Yes], [12, 12, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0012","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9322b015aea93bd3fe33a23759905b2e3241dfa73b11aa10a32c66e43f884b11","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141012,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], target identity=0, and total mapping=[0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(14, 14, [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 13, 13, Yes], [0, 2, 12, 12, Yes], [0, 3, 11, 11, Yes], [0, 4, 10, 10, Yes], [0, 5, 9, 9, Yes], [0, 6, 8, 8, Yes], [0, 7, 7, 7, Yes], [0, 8, 6, 6, Yes], [0, 9, 5, 5, Yes], [0, 10, 4, 4, Yes], [0, 11, 3, 3, Yes], [0, 12, 2, 2, Yes], [0, 13, 1, 1, Yes], [1, 0, 13, 13, Yes], [1, 1, 12, 12, Yes], [1, 2, 11, 11, Yes], [1, 3, 10, 10, Yes], [1, 4, 9, 9, Yes], [1, 5, 8, 8, Yes], [1, 6, 7, 7, Yes], [1, 7, 6, 6, Yes], [1, 8, 5, 5, Yes], [1, 9, 4, 4, Yes], [1, 10, 3, 3, Yes], [1, 11, 2, 2, Yes], [1, 12, 1, 1, Yes], [1, 13, 0, 0, Yes], [2, 0, 12, 12, Yes], [2, 1, 11, 11, Yes], [2, 2, 10, 10, Yes], [2, 3, 9, 9, Yes], [2, 4, 8, 8, Yes], [2, 5, 7, 7, Yes], [2, 6, 6, 6, Yes], [2, 7, 5, 5, Yes], [2, 8, 4, 4, Yes], [2, 9, 3, 3, Yes], [2, 10, 2, 2, Yes], [2, 11, 1, 1, Yes], [2, 12, 0, 0, Yes], [2, 13, 13, 13, Yes], [3, 0, 11, 11, Yes], [3, 1, 10, 10, Yes], [3, 2, 9, 9, Yes], [3, 3, 8, 8, Yes], [3, 4, 7, 7, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [3, 7, 4, 4, Yes], [3, 8, 3, 3, Yes], [3, 9, 2, 2, Yes], [3, 10, 1, 1, Yes], [3, 11, 0, 0, Yes], [3, 12, 13, 13, Yes], [3, 13, 12, 12, Yes], [4, 0, 10, 10, Yes], [4, 1, 9, 9, Yes], [4, 2, 8, 8, Yes], [4, 3, 7, 7, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [4, 7, 3, 3, Yes], [4, 8, 2, 2, Yes], [4, 9, 1, 1, Yes], [4, 10, 0, 0, Yes], [4, 11, 13, 13, Yes], [4, 12, 12, 12, Yes], [4, 13, 11, 11, Yes], [5, 0, 9, 9, Yes], [5, 1, 8, 8, Yes], [5, 2, 7, 7, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [5, 7, 2, 2, Yes], [5, 8, 1, 1, Yes], [5, 9, 0, 0, Yes], [5, 10, 13, 13, Yes], [5, 11, 12, 12, Yes], [5, 12, 11, 11, Yes], [5, 13, 10, 10, Yes], [6, 0, 8, 8, Yes], [6, 1, 7, 7, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes], [6, 7, 1, 1, Yes], [6, 8, 0, 0, Yes], [6, 9, 13, 13, Yes], [6, 10, 12, 12, Yes], [6, 11, 11, 11, Yes], [6, 12, 10, 10, Yes], [6, 13, 9, 9, Yes], [7, 0, 7, 7, Yes], [7, 1, 6, 6, Yes], [7, 2, 5, 5, Yes], [7, 3, 4, 4, Yes], [7, 4, 3, 3, Yes], [7, 5, 2, 2, Yes], [7, 6, 1, 1, Yes], [7, 7, 0, 0, Yes], [7, 8, 13, 13, Yes], [7, 9, 12, 12, Yes], [7, 10, 11, 11, Yes], [7, 11, 10, 10, Yes], [7, 12, 9, 9, Yes], [7, 13, 8, 8, Yes], [8, 0, 6, 6, Yes], [8, 1, 5, 5, Yes], [8, 2, 4, 4, Yes], [8, 3, 3, 3, Yes], [8, 4, 2, 2, Yes], [8, 5, 1, 1, Yes], [8, 6, 0, 0, Yes], [8, 7, 13, 13, Yes], [8, 8, 12, 12, Yes], [8, 9, 11, 11, Yes], [8, 10, 10, 10, Yes], [8, 11, 9, 9, Yes], [8, 12, 8, 8, Yes], [8, 13, 7, 7, Yes], [9, 0, 5, 5, Yes], [9, 1, 4, 4, Yes], [9, 2, 3, 3, Yes], [9, 3, 2, 2, Yes], [9, 4, 1, 1, Yes], [9, 5, 0, 0, Yes], [9, 6, 13, 13, Yes], [9, 7, 12, 12, Yes], [9, 8, 11, 11, Yes], [9, 9, 10, 10, Yes], [9, 10, 9, 9, Yes], [9, 11, 8, 8, Yes], [9, 12, 7, 7, Yes], [9, 13, 6, 6, Yes], [10, 0, 4, 4, Yes], [10, 1, 3, 3, Yes], [10, 2, 2, 2, Yes], [10, 3, 1, 1, Yes], [10, 4, 0, 0, Yes], [10, 5, 13, 13, Yes], [10, 6, 12, 12, Yes], [10, 7, 11, 11, Yes], [10, 8, 10, 10, Yes], [10, 9, 9, 9, Yes], [10, 10, 8, 8, Yes], [10, 11, 7, 7, Yes], [10, 12, 6, 6, Yes], [10, 13, 5, 5, Yes], [11, 0, 3, 3, Yes], [11, 1, 2, 2, Yes], [11, 2, 1, 1, Yes], [11, 3, 0, 0, Yes], [11, 4, 13, 13, Yes], [11, 5, 12, 12, Yes], [11, 6, 11, 11, Yes], [11, 7, 10, 10, Yes], [11, 8, 9, 9, Yes], [11, 9, 8, 8, Yes], [11, 10, 7, 7, Yes], [11, 11, 6, 6, Yes], [11, 12, 5, 5, Yes], [11, 13, 4, 4, Yes], [12, 0, 2, 2, Yes], [12, 1, 1, 1, Yes], [12, 2, 0, 0, Yes], [12, 3, 13, 13, Yes], [12, 4, 12, 12, Yes], [12, 5, 11, 11, Yes], [12, 6, 10, 10, Yes], [12, 7, 9, 9, Yes], [12, 8, 8, 8, Yes], [12, 9, 7, 7, Yes], [12, 10, 6, 6, Yes], [12, 11, 5, 5, Yes], [12, 12, 4, 4, Yes], [12, 13, 3, 3, Yes], [13, 0, 1, 1, Yes], [13, 1, 0, 0, Yes], [13, 2, 13, 13, Yes], [13, 3, 12, 12, Yes], [13, 4, 11, 11, Yes], [13, 5, 10, 10, Yes], [13, 6, 9, 9, Yes], [13, 7, 8, 8, Yes], [13, 8, 7, 7, Yes], [13, 9, 6, 6, Yes], [13, 10, 5, 5, Yes], [13, 11, 4, 4, Yes], [13, 12, 3, 3, Yes], [13, 13, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (14, 14, [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 13, 13, Yes], [0, 2, 12, 12, Yes], [0, 3, 11, 11, Yes], [0, 4, 10, 10, Yes], [0, 5, 9, 9, Yes], [0, 6, 8, 8, Yes], [0, 7, 7, 7, Yes], [0, 8, 6, 6, Yes], [0, 9, 5, 5, Yes], [0, 10, 4, 4, Yes], [0, 11, 3, 3, Yes], [0, 12, 2, 2, Yes], [0, 13, 1, 1, Yes], [1, 0, 13, 13, Yes], [1, 1, 12, 12, Yes], [1, 2, 11, 11, Yes], [1, 3, 10, 10, Yes], [1, 4, 9, 9, Yes], [1, 5, 8, 8, Yes], [1, 6, 7, 7, Yes], [1, 7, 6, 6, Yes], [1, 8, 5, 5, Yes], [1, 9, 4, 4, Yes], [1, 10, 3, 3, Yes], [1, 11, 2, 2, Yes], [1, 12, 1, 1, Yes], [1, 13, 0, 0, Yes], [2, 0, 12, 12, Yes], [2, 1, 11, 11, Yes], [2, 2, 10, 10, Yes], [2, 3, 9, 9, Yes], [2, 4, 8, 8, Yes], [2, 5, 7, 7, Yes], [2, 6, 6, 6, Yes], [2, 7, 5, 5, Yes], [2, 8, 4, 4, Yes], [2, 9, 3, 3, Yes], [2, 10, 2, 2, Yes], [2, 11, 1, 1, Yes], [2, 12, 0, 0, Yes], [2, 13, 13, 13, Yes], [3, 0, 11, 11, Yes], [3, 1, 10, 10, Yes], [3, 2, 9, 9, Yes], [3, 3, 8, 8, Yes], [3, 4, 7, 7, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [3, 7, 4, 4, Yes], [3, 8, 3, 3, Yes], [3, 9, 2, 2, Yes], [3, 10, 1, 1, Yes], [3, 11, 0, 0, Yes], [3, 12, 13, 13, Yes], [3, 13, 12, 12, Yes], [4, 0, 10, 10, Yes], [4, 1, 9, 9, Yes], [4, 2, 8, 8, Yes], [4, 3, 7, 7, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [4, 7, 3, 3, Yes], [4, 8, 2, 2, Yes], [4, 9, 1, 1, Yes], [4, 10, 0, 0, Yes], [4, 11, 13, 13, Yes], [4, 12, 12, 12, Yes], [4, 13, 11, 11, Yes], [5, 0, 9, 9, Yes], [5, 1, 8, 8, Yes], [5, 2, 7, 7, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [5, 7, 2, 2, Yes], [5, 8, 1, 1, Yes], [5, 9, 0, 0, Yes], [5, 10, 13, 13, Yes], [5, 11, 12, 12, Yes], [5, 12, 11, 11, Yes], [5, 13, 10, 10, Yes], [6, 0, 8, 8, Yes], [6, 1, 7, 7, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes], [6, 7, 1, 1, Yes], [6, 8, 0, 0, Yes], [6, 9, 13, 13, Yes], [6, 10, 12, 12, Yes], [6, 11, 11, 11, Yes], [6, 12, 10, 10, Yes], [6, 13, 9, 9, Yes], [7, 0, 7, 7, Yes], [7, 1, 6, 6, Yes], [7, 2, 5, 5, Yes], [7, 3, 4, 4, Yes], [7, 4, 3, 3, Yes], [7, 5, 2, 2, Yes], [7, 6, 1, 1, Yes], [7, 7, 0, 0, Yes], [7, 8, 13, 13, Yes], [7, 9, 12, 12, Yes], [7, 10, 11, 11, Yes], [7, 11, 10, 10, Yes], [7, 12, 9, 9, Yes], [7, 13, 8, 8, Yes], [8, 0, 6, 6, Yes], [8, 1, 5, 5, Yes], [8, 2, 4, 4, Yes], [8, 3, 3, 3, Yes], [8, 4, 2, 2, Yes], [8, 5, 1, 1, Yes], [8, 6, 0, 0, Yes], [8, 7, 13, 13, Yes], [8, 8, 12, 12, Yes], [8, 9, 11, 11, Yes], [8, 10, 10, 10, Yes], [8, 11, 9, 9, Yes], [8, 12, 8, 8, Yes], [8, 13, 7, 7, Yes], [9, 0, 5, 5, Yes], [9, 1, 4, 4, Yes], [9, 2, 3, 3, Yes], [9, 3, 2, 2, Yes], [9, 4, 1, 1, Yes], [9, 5, 0, 0, Yes], [9, 6, 13, 13, Yes], [9, 7, 12, 12, Yes], [9, 8, 11, 11, Yes], [9, 9, 10, 10, Yes], [9, 10, 9, 9, Yes], [9, 11, 8, 8, Yes], [9, 12, 7, 7, Yes], [9, 13, 6, 6, Yes], [10, 0, 4, 4, Yes], [10, 1, 3, 3, Yes], [10, 2, 2, 2, Yes], [10, 3, 1, 1, Yes], [10, 4, 0, 0, Yes], [10, 5, 13, 13, Yes], [10, 6, 12, 12, Yes], [10, 7, 11, 11, Yes], [10, 8, 10, 10, Yes], [10, 9, 9, 9, Yes], [10, 10, 8, 8, Yes], [10, 11, 7, 7, Yes], [10, 12, 6, 6, Yes], [10, 13, 5, 5, Yes], [11, 0, 3, 3, Yes], [11, 1, 2, 2, Yes], [11, 2, 1, 1, Yes], [11, 3, 0, 0, Yes], [11, 4, 13, 13, Yes], [11, 5, 12, 12, Yes], [11, 6, 11, 11, Yes], [11, 7, 10, 10, Yes], [11, 8, 9, 9, Yes], [11, 9, 8, 8, Yes], [11, 10, 7, 7, Yes], [11, 11, 6, 6, Yes], [11, 12, 5, 5, Yes], [11, 13, 4, 4, Yes], [12, 0, 2, 2, Yes], [12, 1, 1, 1, Yes], [12, 2, 0, 0, Yes], [12, 3, 13, 13, Yes], [12, 4, 12, 12, Yes], [12, 5, 11, 11, Yes], [12, 6, 10, 10, Yes], [12, 7, 9, 9, Yes], [12, 8, 8, 8, Yes], [12, 9, 7, 7, Yes], [12, 10, 6, 6, Yes], [12, 11, 5, 5, Yes], [12, 12, 4, 4, Yes], [12, 13, 3, 3, Yes], [13, 0, 1, 1, Yes], [13, 1, 0, 0, Yes], [13, 2, 13, 13, Yes], [13, 3, 12, 12, Yes], [13, 4, 11, 11, Yes], [13, 5, 10, 10, Yes], [13, 6, 9, 9, Yes], [13, 7, 8, 8, Yes], [13, 8, 7, 7, Yes], [13, 9, 6, 6, Yes], [13, 10, 5, 5, Yes], [13, 11, 4, 4, Yes], [13, 12, 3, 3, Yes], [13, 13, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0013","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eaa9d545a11ecf939a939d7ab7f21bdafa6ea16d4d706c373de5b9aa6bc909c","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141013,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], target identity=0, and total mapping=[0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 14, 14, Yes], [0, 2, 13, 13, Yes], [0, 3, 12, 12, Yes], [0, 4, 11, 11, Yes], [0, 5, 10, 10, Yes], [0, 6, 9, 9, Yes], [0, 7, 8, 8, Yes], [0, 8, 7, 7, Yes], [0, 9, 6, 6, Yes], [0, 10, 5, 5, Yes], [0, 11, 4, 4, Yes], [0, 12, 3, 3, Yes], [0, 13, 2, 2, Yes], [0, 14, 1, 1, Yes], [1, 0, 14, 14, Yes], [1, 1, 13, 13, Yes], [1, 2, 12, 12, Yes], [1, 3, 11, 11, Yes], [1, 4, 10, 10, Yes], [1, 5, 9, 9, Yes], [1, 6, 8, 8, Yes], [1, 7, 7, 7, Yes], [1, 8, 6, 6, Yes], [1, 9, 5, 5, Yes], [1, 10, 4, 4, Yes], [1, 11, 3, 3, Yes], [1, 12, 2, 2, Yes], [1, 13, 1, 1, Yes], [1, 14, 0, 0, Yes], [2, 0, 13, 13, Yes], [2, 1, 12, 12, Yes], [2, 2, 11, 11, Yes], [2, 3, 10, 10, Yes], [2, 4, 9, 9, Yes], [2, 5, 8, 8, Yes], [2, 6, 7, 7, Yes], [2, 7, 6, 6, Yes], [2, 8, 5, 5, Yes], [2, 9, 4, 4, Yes], [2, 10, 3, 3, Yes], [2, 11, 2, 2, Yes], [2, 12, 1, 1, Yes], [2, 13, 0, 0, Yes], [2, 14, 14, 14, Yes], [3, 0, 12, 12, Yes], [3, 1, 11, 11, Yes], [3, 2, 10, 10, Yes], [3, 3, 9, 9, Yes], [3, 4, 8, 8, Yes], [3, 5, 7, 7, Yes], [3, 6, 6, 6, Yes], [3, 7, 5, 5, Yes], [3, 8, 4, 4, Yes], [3, 9, 3, 3, Yes], [3, 10, 2, 2, Yes], [3, 11, 1, 1, Yes], [3, 12, 0, 0, Yes], [3, 13, 14, 14, Yes], [3, 14, 13, 13, Yes], [4, 0, 11, 11, Yes], [4, 1, 10, 10, Yes], [4, 2, 9, 9, Yes], [4, 3, 8, 8, Yes], [4, 4, 7, 7, Yes], [4, 5, 6, 6, Yes], [4, 6, 5, 5, Yes], [4, 7, 4, 4, Yes], [4, 8, 3, 3, Yes], [4, 9, 2, 2, Yes], [4, 10, 1, 1, Yes], [4, 11, 0, 0, Yes], [4, 12, 14, 14, Yes], [4, 13, 13, 13, Yes], [4, 14, 12, 12, Yes], [5, 0, 10, 10, Yes], [5, 1, 9, 9, Yes], [5, 2, 8, 8, Yes], [5, 3, 7, 7, Yes], [5, 4, 6, 6, Yes], [5, 5, 5, 5, Yes], [5, 6, 4, 4, Yes], [5, 7, 3, 3, Yes], [5, 8, 2, 2, Yes], [5, 9, 1, 1, Yes], [5, 10, 0, 0, Yes], [5, 11, 14, 14, Yes], [5, 12, 13, 13, Yes], [5, 13, 12, 12, Yes], [5, 14, 11, 11, Yes], [6, 0, 9, 9, Yes], [6, 1, 8, 8, Yes], [6, 2, 7, 7, Yes], [6, 3, 6, 6, Yes], [6, 4, 5, 5, Yes], [6, 5, 4, 4, Yes], [6, 6, 3, 3, Yes], [6, 7, 2, 2, Yes], [6, 8, 1, 1, Yes], [6, 9, 0, 0, Yes], [6, 10, 14, 14, Yes], [6, 11, 13, 13, Yes], [6, 12, 12, 12, Yes], [6, 13, 11, 11, Yes], [6, 14, 10, 10, Yes], [7, 0, 8, 8, Yes], [7, 1, 7, 7, Yes], [7, 2, 6, 6, Yes], [7, 3, 5, 5, Yes], [7, 4, 4, 4, Yes], [7, 5, 3, 3, Yes], [7, 6, 2, 2, Yes], [7, 7, 1, 1, Yes], [7, 8, 0, 0, Yes], [7, 9, 14, 14, Yes], [7, 10, 13, 13, Yes], [7, 11, 12, 12, Yes], [7, 12, 11, 11, Yes], [7, 13, 10, 10, Yes], [7, 14, 9, 9, Yes], [8, 0, 7, 7, Yes], [8, 1, 6, 6, Yes], [8, 2, 5, 5, Yes], [8, 3, 4, 4, Yes], [8, 4, 3, 3, Yes], [8, 5, 2, 2, Yes], [8, 6, 1, 1, Yes], [8, 7, 0, 0, Yes], [8, 8, 14, 14, Yes], [8, 9, 13, 13, Yes], [8, 10, 12, 12, Yes], [8, 11, 11, 11, Yes], [8, 12, 10, 10, Yes], [8, 13, 9, 9, Yes], [8, 14, 8, 8, Yes], [9, 0, 6, 6, Yes], [9, 1, 5, 5, Yes], [9, 2, 4, 4, Yes], [9, 3, 3, 3, Yes], [9, 4, 2, 2, Yes], [9, 5, 1, 1, Yes], [9, 6, 0, 0, Yes], [9, 7, 14, 14, Yes], [9, 8, 13, 13, Yes], [9, 9, 12, 12, Yes], [9, 10, 11, 11, Yes], [9, 11, 10, 10, Yes], [9, 12, 9, 9, Yes], [9, 13, 8, 8, Yes], [9, 14, 7, 7, Yes], [10, 0, 5, 5, Yes], [10, 1, 4, 4, Yes], [10, 2, 3, 3, Yes], [10, 3, 2, 2, Yes], [10, 4, 1, 1, Yes], [10, 5, 0, 0, Yes], [10, 6, 14, 14, Yes], [10, 7, 13, 13, Yes], [10, 8, 12, 12, Yes], [10, 9, 11, 11, Yes], [10, 10, 10, 10, Yes], [10, 11, 9, 9, Yes], [10, 12, 8, 8, Yes], [10, 13, 7, 7, Yes], [10, 14, 6, 6, Yes], [11, 0, 4, 4, Yes], [11, 1, 3, 3, Yes], [11, 2, 2, 2, Yes], [11, 3, 1, 1, Yes], [11, 4, 0, 0, Yes], [11, 5, 14, 14, Yes], [11, 6, 13, 13, Yes], [11, 7, 12, 12, Yes], [11, 8, 11, 11, Yes], [11, 9, 10, 10, Yes], [11, 10, 9, 9, Yes], [11, 11, 8, 8, Yes], [11, 12, 7, 7, Yes], [11, 13, 6, 6, Yes], [11, 14, 5, 5, Yes], [12, 0, 3, 3, Yes], [12, 1, 2, 2, Yes], [12, 2, 1, 1, Yes], [12, 3, 0, 0, Yes], [12, 4, 14, 14, Yes], [12, 5, 13, 13, Yes], [12, 6, 12, 12, Yes], [12, 7, 11, 11, Yes], [12, 8, 10, 10, Yes], [12, 9, 9, 9, Yes], [12, 10, 8, 8, Yes], [12, 11, 7, 7, Yes], [12, 12, 6, 6, Yes], [12, 13, 5, 5, Yes], [12, 14, 4, 4, Yes], [13, 0, 2, 2, Yes], [13, 1, 1, 1, Yes], [13, 2, 0, 0, Yes], [13, 3, 14, 14, Yes], [13, 4, 13, 13, Yes], [13, 5, 12, 12, Yes], [13, 6, 11, 11, Yes], [13, 7, 10, 10, Yes], [13, 8, 9, 9, Yes], [13, 9, 8, 8, Yes], [13, 10, 7, 7, Yes], [13, 11, 6, 6, Yes], [13, 12, 5, 5, Yes], [13, 13, 4, 4, Yes], [13, 14, 3, 3, Yes], [14, 0, 1, 1, Yes], [14, 1, 0, 0, Yes], [14, 2, 14, 14, Yes], [14, 3, 13, 13, Yes], [14, 4, 12, 12, Yes], [14, 5, 11, 11, Yes], [14, 6, 10, 10, Yes], [14, 7, 9, 9, Yes], [14, 8, 8, 8, Yes], [14, 9, 7, 7, Yes], [14, 10, 6, 6, Yes], [14, 11, 5, 5, Yes], [14, 12, 4, 4, Yes], [14, 13, 3, 3, Yes], [14, 14, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 14, 14, Yes], [0, 2, 13, 13, Yes], [0, 3, 12, 12, Yes], [0, 4, 11, 11, Yes], [0, 5, 10, 10, Yes], [0, 6, 9, 9, Yes], [0, 7, 8, 8, Yes], [0, 8, 7, 7, Yes], [0, 9, 6, 6, Yes], [0, 10, 5, 5, Yes], [0, 11, 4, 4, Yes], [0, 12, 3, 3, Yes], [0, 13, 2, 2, Yes], [0, 14, 1, 1, Yes], [1, 0, 14, 14, Yes], [1, 1, 13, 13, Yes], [1, 2, 12, 12, Yes], [1, 3, 11, 11, Yes], [1, 4, 10, 10, Yes], [1, 5, 9, 9, Yes], [1, 6, 8, 8, Yes], [1, 7, 7, 7, Yes], [1, 8, 6, 6, Yes], [1, 9, 5, 5, Yes], [1, 10, 4, 4, Yes], [1, 11, 3, 3, Yes], [1, 12, 2, 2, Yes], [1, 13, 1, 1, Yes], [1, 14, 0, 0, Yes], [2, 0, 13, 13, Yes], [2, 1, 12, 12, Yes], [2, 2, 11, 11, Yes], [2, 3, 10, 10, Yes], [2, 4, 9, 9, Yes], [2, 5, 8, 8, Yes], [2, 6, 7, 7, Yes], [2, 7, 6, 6, Yes], [2, 8, 5, 5, Yes], [2, 9, 4, 4, Yes], [2, 10, 3, 3, Yes], [2, 11, 2, 2, Yes], [2, 12, 1, 1, Yes], [2, 13, 0, 0, Yes], [2, 14, 14, 14, Yes], [3, 0, 12, 12, Yes], [3, 1, 11, 11, Yes], [3, 2, 10, 10, Yes], [3, 3, 9, 9, Yes], [3, 4, 8, 8, Yes], [3, 5, 7, 7, Yes], [3, 6, 6, 6, Yes], [3, 7, 5, 5, Yes], [3, 8, 4, 4, Yes], [3, 9, 3, 3, Yes], [3, 10, 2, 2, Yes], [3, 11, 1, 1, Yes], [3, 12, 0, 0, Yes], [3, 13, 14, 14, Yes], [3, 14, 13, 13, Yes], [4, 0, 11, 11, Yes], [4, 1, 10, 10, Yes], [4, 2, 9, 9, Yes], [4, 3, 8, 8, Yes], [4, 4, 7, 7, Yes], [4, 5, 6, 6, Yes], [4, 6, 5, 5, Yes], [4, 7, 4, 4, Yes], [4, 8, 3, 3, Yes], [4, 9, 2, 2, Yes], [4, 10, 1, 1, Yes], [4, 11, 0, 0, Yes], [4, 12, 14, 14, Yes], [4, 13, 13, 13, Yes], [4, 14, 12, 12, Yes], [5, 0, 10, 10, Yes], [5, 1, 9, 9, Yes], [5, 2, 8, 8, Yes], [5, 3, 7, 7, Yes], [5, 4, 6, 6, Yes], [5, 5, 5, 5, Yes], [5, 6, 4, 4, Yes], [5, 7, 3, 3, Yes], [5, 8, 2, 2, Yes], [5, 9, 1, 1, Yes], [5, 10, 0, 0, Yes], [5, 11, 14, 14, Yes], [5, 12, 13, 13, Yes], [5, 13, 12, 12, Yes], [5, 14, 11, 11, Yes], [6, 0, 9, 9, Yes], [6, 1, 8, 8, Yes], [6, 2, 7, 7, Yes], [6, 3, 6, 6, Yes], [6, 4, 5, 5, Yes], [6, 5, 4, 4, Yes], [6, 6, 3, 3, Yes], [6, 7, 2, 2, Yes], [6, 8, 1, 1, Yes], [6, 9, 0, 0, Yes], [6, 10, 14, 14, Yes], [6, 11, 13, 13, Yes], [6, 12, 12, 12, Yes], [6, 13, 11, 11, Yes], [6, 14, 10, 10, Yes], [7, 0, 8, 8, Yes], [7, 1, 7, 7, Yes], [7, 2, 6, 6, Yes], [7, 3, 5, 5, Yes], [7, 4, 4, 4, Yes], [7, 5, 3, 3, Yes], [7, 6, 2, 2, Yes], [7, 7, 1, 1, Yes], [7, 8, 0, 0, Yes], [7, 9, 14, 14, Yes], [7, 10, 13, 13, Yes], [7, 11, 12, 12, Yes], [7, 12, 11, 11, Yes], [7, 13, 10, 10, Yes], [7, 14, 9, 9, Yes], [8, 0, 7, 7, Yes], [8, 1, 6, 6, Yes], [8, 2, 5, 5, Yes], [8, 3, 4, 4, Yes], [8, 4, 3, 3, Yes], [8, 5, 2, 2, Yes], [8, 6, 1, 1, Yes], [8, 7, 0, 0, Yes], [8, 8, 14, 14, Yes], [8, 9, 13, 13, Yes], [8, 10, 12, 12, Yes], [8, 11, 11, 11, Yes], [8, 12, 10, 10, Yes], [8, 13, 9, 9, Yes], [8, 14, 8, 8, Yes], [9, 0, 6, 6, Yes], [9, 1, 5, 5, Yes], [9, 2, 4, 4, Yes], [9, 3, 3, 3, Yes], [9, 4, 2, 2, Yes], [9, 5, 1, 1, Yes], [9, 6, 0, 0, Yes], [9, 7, 14, 14, Yes], [9, 8, 13, 13, Yes], [9, 9, 12, 12, Yes], [9, 10, 11, 11, Yes], [9, 11, 10, 10, Yes], [9, 12, 9, 9, Yes], [9, 13, 8, 8, Yes], [9, 14, 7, 7, Yes], [10, 0, 5, 5, Yes], [10, 1, 4, 4, Yes], [10, 2, 3, 3, Yes], [10, 3, 2, 2, Yes], [10, 4, 1, 1, Yes], [10, 5, 0, 0, Yes], [10, 6, 14, 14, Yes], [10, 7, 13, 13, Yes], [10, 8, 12, 12, Yes], [10, 9, 11, 11, Yes], [10, 10, 10, 10, Yes], [10, 11, 9, 9, Yes], [10, 12, 8, 8, Yes], [10, 13, 7, 7, Yes], [10, 14, 6, 6, Yes], [11, 0, 4, 4, Yes], [11, 1, 3, 3, Yes], [11, 2, 2, 2, Yes], [11, 3, 1, 1, Yes], [11, 4, 0, 0, Yes], [11, 5, 14, 14, Yes], [11, 6, 13, 13, Yes], [11, 7, 12, 12, Yes], [11, 8, 11, 11, Yes], [11, 9, 10, 10, Yes], [11, 10, 9, 9, Yes], [11, 11, 8, 8, Yes], [11, 12, 7, 7, Yes], [11, 13, 6, 6, Yes], [11, 14, 5, 5, Yes], [12, 0, 3, 3, Yes], [12, 1, 2, 2, Yes], [12, 2, 1, 1, Yes], [12, 3, 0, 0, Yes], [12, 4, 14, 14, Yes], [12, 5, 13, 13, Yes], [12, 6, 12, 12, Yes], [12, 7, 11, 11, Yes], [12, 8, 10, 10, Yes], [12, 9, 9, 9, Yes], [12, 10, 8, 8, Yes], [12, 11, 7, 7, Yes], [12, 12, 6, 6, Yes], [12, 13, 5, 5, Yes], [12, 14, 4, 4, Yes], [13, 0, 2, 2, Yes], [13, 1, 1, 1, Yes], [13, 2, 0, 0, Yes], [13, 3, 14, 14, Yes], [13, 4, 13, 13, Yes], [13, 5, 12, 12, Yes], [13, 6, 11, 11, Yes], [13, 7, 10, 10, Yes], [13, 8, 9, 9, Yes], [13, 9, 8, 8, Yes], [13, 10, 7, 7, Yes], [13, 11, 6, 6, Yes], [13, 12, 5, 5, Yes], [13, 13, 4, 4, Yes], [13, 14, 3, 3, Yes], [14, 0, 1, 1, Yes], [14, 1, 0, 0, Yes], [14, 2, 14, 14, Yes], [14, 3, 13, 13, Yes], [14, 4, 12, 12, Yes], [14, 5, 11, 11, Yes], [14, 6, 10, 10, Yes], [14, 7, 9, 9, Yes], [14, 8, 8, 8, Yes], [14, 9, 7, 7, Yes], [14, 10, 6, 6, Yes], [14, 11, 5, 5, Yes], [14, 12, 4, 4, Yes], [14, 13, 3, 3, Yes], [14, 14, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0014","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f5c3675bb3cf03c94290022a0ce121a9f7b67a0138c651a7f77df4ddbc9d6c4","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141014,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], target identity=0, and total mapping=[0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(16, 16, [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 15, 15, Yes], [0, 2, 14, 14, Yes], [0, 3, 13, 13, Yes], [0, 4, 12, 12, Yes], [0, 5, 11, 11, Yes], [0, 6, 10, 10, Yes], [0, 7, 9, 9, Yes], [0, 8, 8, 8, Yes], [0, 9, 7, 7, Yes], [0, 10, 6, 6, Yes], [0, 11, 5, 5, Yes], [0, 12, 4, 4, Yes], [0, 13, 3, 3, Yes], [0, 14, 2, 2, Yes], [0, 15, 1, 1, Yes], [1, 0, 15, 15, Yes], [1, 1, 14, 14, Yes], [1, 2, 13, 13, Yes], [1, 3, 12, 12, Yes], [1, 4, 11, 11, Yes], [1, 5, 10, 10, Yes], [1, 6, 9, 9, Yes], [1, 7, 8, 8, Yes], [1, 8, 7, 7, Yes], [1, 9, 6, 6, Yes], [1, 10, 5, 5, Yes], [1, 11, 4, 4, Yes], [1, 12, 3, 3, Yes], [1, 13, 2, 2, Yes], [1, 14, 1, 1, Yes], [1, 15, 0, 0, Yes], [2, 0, 14, 14, Yes], [2, 1, 13, 13, Yes], [2, 2, 12, 12, Yes], [2, 3, 11, 11, Yes], [2, 4, 10, 10, Yes], [2, 5, 9, 9, Yes], [2, 6, 8, 8, Yes], [2, 7, 7, 7, Yes], [2, 8, 6, 6, Yes], [2, 9, 5, 5, Yes], [2, 10, 4, 4, Yes], [2, 11, 3, 3, Yes], [2, 12, 2, 2, Yes], [2, 13, 1, 1, Yes], [2, 14, 0, 0, Yes], [2, 15, 15, 15, Yes], [3, 0, 13, 13, Yes], [3, 1, 12, 12, Yes], [3, 2, 11, 11, Yes], [3, 3, 10, 10, Yes], [3, 4, 9, 9, Yes], [3, 5, 8, 8, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [3, 8, 5, 5, Yes], [3, 9, 4, 4, Yes], [3, 10, 3, 3, Yes], [3, 11, 2, 2, Yes], [3, 12, 1, 1, Yes], [3, 13, 0, 0, Yes], [3, 14, 15, 15, Yes], [3, 15, 14, 14, Yes], [4, 0, 12, 12, Yes], [4, 1, 11, 11, Yes], [4, 2, 10, 10, Yes], [4, 3, 9, 9, Yes], [4, 4, 8, 8, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [4, 8, 4, 4, Yes], [4, 9, 3, 3, Yes], [4, 10, 2, 2, Yes], [4, 11, 1, 1, Yes], [4, 12, 0, 0, Yes], [4, 13, 15, 15, Yes], [4, 14, 14, 14, Yes], [4, 15, 13, 13, Yes], [5, 0, 11, 11, Yes], [5, 1, 10, 10, Yes], [5, 2, 9, 9, Yes], [5, 3, 8, 8, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [5, 8, 3, 3, Yes], [5, 9, 2, 2, Yes], [5, 10, 1, 1, Yes], [5, 11, 0, 0, Yes], [5, 12, 15, 15, Yes], [5, 13, 14, 14, Yes], [5, 14, 13, 13, Yes], [5, 15, 12, 12, Yes], [6, 0, 10, 10, Yes], [6, 1, 9, 9, Yes], [6, 2, 8, 8, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [6, 8, 2, 2, Yes], [6, 9, 1, 1, Yes], [6, 10, 0, 0, Yes], [6, 11, 15, 15, Yes], [6, 12, 14, 14, Yes], [6, 13, 13, 13, Yes], [6, 14, 12, 12, Yes], [6, 15, 11, 11, Yes], [7, 0, 9, 9, Yes], [7, 1, 8, 8, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes], [7, 8, 1, 1, Yes], [7, 9, 0, 0, Yes], [7, 10, 15, 15, Yes], [7, 11, 14, 14, Yes], [7, 12, 13, 13, Yes], [7, 13, 12, 12, Yes], [7, 14, 11, 11, Yes], [7, 15, 10, 10, Yes], [8, 0, 8, 8, Yes], [8, 1, 7, 7, Yes], [8, 2, 6, 6, Yes], [8, 3, 5, 5, Yes], [8, 4, 4, 4, Yes], [8, 5, 3, 3, Yes], [8, 6, 2, 2, Yes], [8, 7, 1, 1, Yes], [8, 8, 0, 0, Yes], [8, 9, 15, 15, Yes], [8, 10, 14, 14, Yes], [8, 11, 13, 13, Yes], [8, 12, 12, 12, Yes], [8, 13, 11, 11, Yes], [8, 14, 10, 10, Yes], [8, 15, 9, 9, Yes], [9, 0, 7, 7, Yes], [9, 1, 6, 6, Yes], [9, 2, 5, 5, Yes], [9, 3, 4, 4, Yes], [9, 4, 3, 3, Yes], [9, 5, 2, 2, Yes], [9, 6, 1, 1, Yes], [9, 7, 0, 0, Yes], [9, 8, 15, 15, Yes], [9, 9, 14, 14, Yes], [9, 10, 13, 13, Yes], [9, 11, 12, 12, Yes], [9, 12, 11, 11, Yes], [9, 13, 10, 10, Yes], [9, 14, 9, 9, Yes], [9, 15, 8, 8, Yes], [10, 0, 6, 6, Yes], [10, 1, 5, 5, Yes], [10, 2, 4, 4, Yes], [10, 3, 3, 3, Yes], [10, 4, 2, 2, Yes], [10, 5, 1, 1, Yes], [10, 6, 0, 0, Yes], [10, 7, 15, 15, Yes], [10, 8, 14, 14, Yes], [10, 9, 13, 13, Yes], [10, 10, 12, 12, Yes], [10, 11, 11, 11, Yes], [10, 12, 10, 10, Yes], [10, 13, 9, 9, Yes], [10, 14, 8, 8, Yes], [10, 15, 7, 7, Yes], [11, 0, 5, 5, Yes], [11, 1, 4, 4, Yes], [11, 2, 3, 3, Yes], [11, 3, 2, 2, Yes], [11, 4, 1, 1, Yes], [11, 5, 0, 0, Yes], [11, 6, 15, 15, Yes], [11, 7, 14, 14, Yes], [11, 8, 13, 13, Yes], [11, 9, 12, 12, Yes], [11, 10, 11, 11, Yes], [11, 11, 10, 10, Yes], [11, 12, 9, 9, Yes], [11, 13, 8, 8, Yes], [11, 14, 7, 7, Yes], [11, 15, 6, 6, Yes], [12, 0, 4, 4, Yes], [12, 1, 3, 3, Yes], [12, 2, 2, 2, Yes], [12, 3, 1, 1, Yes], [12, 4, 0, 0, Yes], [12, 5, 15, 15, Yes], [12, 6, 14, 14, Yes], [12, 7, 13, 13, Yes], [12, 8, 12, 12, Yes], [12, 9, 11, 11, Yes], [12, 10, 10, 10, Yes], [12, 11, 9, 9, Yes], [12, 12, 8, 8, Yes], [12, 13, 7, 7, Yes], [12, 14, 6, 6, Yes], [12, 15, 5, 5, Yes], [13, 0, 3, 3, Yes], [13, 1, 2, 2, Yes], [13, 2, 1, 1, Yes], [13, 3, 0, 0, Yes], [13, 4, 15, 15, Yes], [13, 5, 14, 14, Yes], [13, 6, 13, 13, Yes], [13, 7, 12, 12, Yes], [13, 8, 11, 11, Yes], [13, 9, 10, 10, Yes], [13, 10, 9, 9, Yes], [13, 11, 8, 8, Yes], [13, 12, 7, 7, Yes], [13, 13, 6, 6, Yes], [13, 14, 5, 5, Yes], [13, 15, 4, 4, Yes], [14, 0, 2, 2, Yes], [14, 1, 1, 1, Yes], [14, 2, 0, 0, Yes], [14, 3, 15, 15, Yes], [14, 4, 14, 14, Yes], [14, 5, 13, 13, Yes], [14, 6, 12, 12, Yes], [14, 7, 11, 11, Yes], [14, 8, 10, 10, Yes], [14, 9, 9, 9, Yes], [14, 10, 8, 8, Yes], [14, 11, 7, 7, Yes], [14, 12, 6, 6, Yes], [14, 13, 5, 5, Yes], [14, 14, 4, 4, Yes], [14, 15, 3, 3, Yes], [15, 0, 1, 1, Yes], [15, 1, 0, 0, Yes], [15, 2, 15, 15, Yes], [15, 3, 14, 14, Yes], [15, 4, 13, 13, Yes], [15, 5, 12, 12, Yes], [15, 6, 11, 11, Yes], [15, 7, 10, 10, Yes], [15, 8, 9, 9, Yes], [15, 9, 8, 8, Yes], [15, 10, 7, 7, Yes], [15, 11, 6, 6, Yes], [15, 12, 5, 5, Yes], [15, 13, 4, 4, Yes], [15, 14, 3, 3, Yes], [15, 15, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (16, 16, [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 15, 15, Yes], [0, 2, 14, 14, Yes], [0, 3, 13, 13, Yes], [0, 4, 12, 12, Yes], [0, 5, 11, 11, Yes], [0, 6, 10, 10, Yes], [0, 7, 9, 9, Yes], [0, 8, 8, 8, Yes], [0, 9, 7, 7, Yes], [0, 10, 6, 6, Yes], [0, 11, 5, 5, Yes], [0, 12, 4, 4, Yes], [0, 13, 3, 3, Yes], [0, 14, 2, 2, Yes], [0, 15, 1, 1, Yes], [1, 0, 15, 15, Yes], [1, 1, 14, 14, Yes], [1, 2, 13, 13, Yes], [1, 3, 12, 12, Yes], [1, 4, 11, 11, Yes], [1, 5, 10, 10, Yes], [1, 6, 9, 9, Yes], [1, 7, 8, 8, Yes], [1, 8, 7, 7, Yes], [1, 9, 6, 6, Yes], [1, 10, 5, 5, Yes], [1, 11, 4, 4, Yes], [1, 12, 3, 3, Yes], [1, 13, 2, 2, Yes], [1, 14, 1, 1, Yes], [1, 15, 0, 0, Yes], [2, 0, 14, 14, Yes], [2, 1, 13, 13, Yes], [2, 2, 12, 12, Yes], [2, 3, 11, 11, Yes], [2, 4, 10, 10, Yes], [2, 5, 9, 9, Yes], [2, 6, 8, 8, Yes], [2, 7, 7, 7, Yes], [2, 8, 6, 6, Yes], [2, 9, 5, 5, Yes], [2, 10, 4, 4, Yes], [2, 11, 3, 3, Yes], [2, 12, 2, 2, Yes], [2, 13, 1, 1, Yes], [2, 14, 0, 0, Yes], [2, 15, 15, 15, Yes], [3, 0, 13, 13, Yes], [3, 1, 12, 12, Yes], [3, 2, 11, 11, Yes], [3, 3, 10, 10, Yes], [3, 4, 9, 9, Yes], [3, 5, 8, 8, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [3, 8, 5, 5, Yes], [3, 9, 4, 4, Yes], [3, 10, 3, 3, Yes], [3, 11, 2, 2, Yes], [3, 12, 1, 1, Yes], [3, 13, 0, 0, Yes], [3, 14, 15, 15, Yes], [3, 15, 14, 14, Yes], [4, 0, 12, 12, Yes], [4, 1, 11, 11, Yes], [4, 2, 10, 10, Yes], [4, 3, 9, 9, Yes], [4, 4, 8, 8, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [4, 8, 4, 4, Yes], [4, 9, 3, 3, Yes], [4, 10, 2, 2, Yes], [4, 11, 1, 1, Yes], [4, 12, 0, 0, Yes], [4, 13, 15, 15, Yes], [4, 14, 14, 14, Yes], [4, 15, 13, 13, Yes], [5, 0, 11, 11, Yes], [5, 1, 10, 10, Yes], [5, 2, 9, 9, Yes], [5, 3, 8, 8, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [5, 8, 3, 3, Yes], [5, 9, 2, 2, Yes], [5, 10, 1, 1, Yes], [5, 11, 0, 0, Yes], [5, 12, 15, 15, Yes], [5, 13, 14, 14, Yes], [5, 14, 13, 13, Yes], [5, 15, 12, 12, Yes], [6, 0, 10, 10, Yes], [6, 1, 9, 9, Yes], [6, 2, 8, 8, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [6, 8, 2, 2, Yes], [6, 9, 1, 1, Yes], [6, 10, 0, 0, Yes], [6, 11, 15, 15, Yes], [6, 12, 14, 14, Yes], [6, 13, 13, 13, Yes], [6, 14, 12, 12, Yes], [6, 15, 11, 11, Yes], [7, 0, 9, 9, Yes], [7, 1, 8, 8, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes], [7, 8, 1, 1, Yes], [7, 9, 0, 0, Yes], [7, 10, 15, 15, Yes], [7, 11, 14, 14, Yes], [7, 12, 13, 13, Yes], [7, 13, 12, 12, Yes], [7, 14, 11, 11, Yes], [7, 15, 10, 10, Yes], [8, 0, 8, 8, Yes], [8, 1, 7, 7, Yes], [8, 2, 6, 6, Yes], [8, 3, 5, 5, Yes], [8, 4, 4, 4, Yes], [8, 5, 3, 3, Yes], [8, 6, 2, 2, Yes], [8, 7, 1, 1, Yes], [8, 8, 0, 0, Yes], [8, 9, 15, 15, Yes], [8, 10, 14, 14, Yes], [8, 11, 13, 13, Yes], [8, 12, 12, 12, Yes], [8, 13, 11, 11, Yes], [8, 14, 10, 10, Yes], [8, 15, 9, 9, Yes], [9, 0, 7, 7, Yes], [9, 1, 6, 6, Yes], [9, 2, 5, 5, Yes], [9, 3, 4, 4, Yes], [9, 4, 3, 3, Yes], [9, 5, 2, 2, Yes], [9, 6, 1, 1, Yes], [9, 7, 0, 0, Yes], [9, 8, 15, 15, Yes], [9, 9, 14, 14, Yes], [9, 10, 13, 13, Yes], [9, 11, 12, 12, Yes], [9, 12, 11, 11, Yes], [9, 13, 10, 10, Yes], [9, 14, 9, 9, Yes], [9, 15, 8, 8, Yes], [10, 0, 6, 6, Yes], [10, 1, 5, 5, Yes], [10, 2, 4, 4, Yes], [10, 3, 3, 3, Yes], [10, 4, 2, 2, Yes], [10, 5, 1, 1, Yes], [10, 6, 0, 0, Yes], [10, 7, 15, 15, Yes], [10, 8, 14, 14, Yes], [10, 9, 13, 13, Yes], [10, 10, 12, 12, Yes], [10, 11, 11, 11, Yes], [10, 12, 10, 10, Yes], [10, 13, 9, 9, Yes], [10, 14, 8, 8, Yes], [10, 15, 7, 7, Yes], [11, 0, 5, 5, Yes], [11, 1, 4, 4, Yes], [11, 2, 3, 3, Yes], [11, 3, 2, 2, Yes], [11, 4, 1, 1, Yes], [11, 5, 0, 0, Yes], [11, 6, 15, 15, Yes], [11, 7, 14, 14, Yes], [11, 8, 13, 13, Yes], [11, 9, 12, 12, Yes], [11, 10, 11, 11, Yes], [11, 11, 10, 10, Yes], [11, 12, 9, 9, Yes], [11, 13, 8, 8, Yes], [11, 14, 7, 7, Yes], [11, 15, 6, 6, Yes], [12, 0, 4, 4, Yes], [12, 1, 3, 3, Yes], [12, 2, 2, 2, Yes], [12, 3, 1, 1, Yes], [12, 4, 0, 0, Yes], [12, 5, 15, 15, Yes], [12, 6, 14, 14, Yes], [12, 7, 13, 13, Yes], [12, 8, 12, 12, Yes], [12, 9, 11, 11, Yes], [12, 10, 10, 10, Yes], [12, 11, 9, 9, Yes], [12, 12, 8, 8, Yes], [12, 13, 7, 7, Yes], [12, 14, 6, 6, Yes], [12, 15, 5, 5, Yes], [13, 0, 3, 3, Yes], [13, 1, 2, 2, Yes], [13, 2, 1, 1, Yes], [13, 3, 0, 0, Yes], [13, 4, 15, 15, Yes], [13, 5, 14, 14, Yes], [13, 6, 13, 13, Yes], [13, 7, 12, 12, Yes], [13, 8, 11, 11, Yes], [13, 9, 10, 10, Yes], [13, 10, 9, 9, Yes], [13, 11, 8, 8, Yes], [13, 12, 7, 7, Yes], [13, 13, 6, 6, Yes], [13, 14, 5, 5, Yes], [13, 15, 4, 4, Yes], [14, 0, 2, 2, Yes], [14, 1, 1, 1, Yes], [14, 2, 0, 0, Yes], [14, 3, 15, 15, Yes], [14, 4, 14, 14, Yes], [14, 5, 13, 13, Yes], [14, 6, 12, 12, Yes], [14, 7, 11, 11, Yes], [14, 8, 10, 10, Yes], [14, 9, 9, 9, Yes], [14, 10, 8, 8, Yes], [14, 11, 7, 7, Yes], [14, 12, 6, 6, Yes], [14, 13, 5, 5, Yes], [14, 14, 4, 4, Yes], [14, 15, 3, 3, Yes], [15, 0, 1, 1, Yes], [15, 1, 0, 0, Yes], [15, 2, 15, 15, Yes], [15, 3, 14, 14, Yes], [15, 4, 13, 13, Yes], [15, 5, 12, 12, Yes], [15, 6, 11, 11, Yes], [15, 7, 10, 10, Yes], [15, 8, 9, 9, Yes], [15, 9, 8, 8, Yes], [15, 10, 7, 7, Yes], [15, 11, 6, 6, Yes], [15, 12, 5, 5, Yes], [15, 13, 4, 4, Yes], [15, 14, 3, 3, Yes], [15, 15, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0015","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7401f8dce4c817d9744f95a94fac8db55f68c6df25a77fb3192bfa5555df551c","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141015,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], target identity=0, and total mapping=[0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(17, 17, [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 16, 16, Yes], [0, 2, 15, 15, Yes], [0, 3, 14, 14, Yes], [0, 4, 13, 13, Yes], [0, 5, 12, 12, Yes], [0, 6, 11, 11, Yes], [0, 7, 10, 10, Yes], [0, 8, 9, 9, Yes], [0, 9, 8, 8, Yes], [0, 10, 7, 7, Yes], [0, 11, 6, 6, Yes], [0, 12, 5, 5, Yes], [0, 13, 4, 4, Yes], [0, 14, 3, 3, Yes], [0, 15, 2, 2, Yes], [0, 16, 1, 1, Yes], [1, 0, 16, 16, Yes], [1, 1, 15, 15, Yes], [1, 2, 14, 14, Yes], [1, 3, 13, 13, Yes], [1, 4, 12, 12, Yes], [1, 5, 11, 11, Yes], [1, 6, 10, 10, Yes], [1, 7, 9, 9, Yes], [1, 8, 8, 8, Yes], [1, 9, 7, 7, Yes], [1, 10, 6, 6, Yes], [1, 11, 5, 5, Yes], [1, 12, 4, 4, Yes], [1, 13, 3, 3, Yes], [1, 14, 2, 2, Yes], [1, 15, 1, 1, Yes], [1, 16, 0, 0, Yes], [2, 0, 15, 15, Yes], [2, 1, 14, 14, Yes], [2, 2, 13, 13, Yes], [2, 3, 12, 12, Yes], [2, 4, 11, 11, Yes], [2, 5, 10, 10, Yes], [2, 6, 9, 9, Yes], [2, 7, 8, 8, Yes], [2, 8, 7, 7, Yes], [2, 9, 6, 6, Yes], [2, 10, 5, 5, Yes], [2, 11, 4, 4, Yes], [2, 12, 3, 3, Yes], [2, 13, 2, 2, Yes], [2, 14, 1, 1, Yes], [2, 15, 0, 0, Yes], [2, 16, 16, 16, Yes], [3, 0, 14, 14, Yes], [3, 1, 13, 13, Yes], [3, 2, 12, 12, Yes], [3, 3, 11, 11, Yes], [3, 4, 10, 10, Yes], [3, 5, 9, 9, Yes], [3, 6, 8, 8, Yes], [3, 7, 7, 7, Yes], [3, 8, 6, 6, Yes], [3, 9, 5, 5, Yes], [3, 10, 4, 4, Yes], [3, 11, 3, 3, Yes], [3, 12, 2, 2, Yes], [3, 13, 1, 1, Yes], [3, 14, 0, 0, Yes], [3, 15, 16, 16, Yes], [3, 16, 15, 15, Yes], [4, 0, 13, 13, Yes], [4, 1, 12, 12, Yes], [4, 2, 11, 11, Yes], [4, 3, 10, 10, Yes], [4, 4, 9, 9, Yes], [4, 5, 8, 8, Yes], [4, 6, 7, 7, Yes], [4, 7, 6, 6, Yes], [4, 8, 5, 5, Yes], [4, 9, 4, 4, Yes], [4, 10, 3, 3, Yes], [4, 11, 2, 2, Yes], [4, 12, 1, 1, Yes], [4, 13, 0, 0, Yes], [4, 14, 16, 16, Yes], [4, 15, 15, 15, Yes], [4, 16, 14, 14, Yes], [5, 0, 12, 12, Yes], [5, 1, 11, 11, Yes], [5, 2, 10, 10, Yes], [5, 3, 9, 9, Yes], [5, 4, 8, 8, Yes], [5, 5, 7, 7, Yes], [5, 6, 6, 6, Yes], [5, 7, 5, 5, Yes], [5, 8, 4, 4, Yes], [5, 9, 3, 3, Yes], [5, 10, 2, 2, Yes], [5, 11, 1, 1, Yes], [5, 12, 0, 0, Yes], [5, 13, 16, 16, Yes], [5, 14, 15, 15, Yes], [5, 15, 14, 14, Yes], [5, 16, 13, 13, Yes], [6, 0, 11, 11, Yes], [6, 1, 10, 10, Yes], [6, 2, 9, 9, Yes], [6, 3, 8, 8, Yes], [6, 4, 7, 7, Yes], [6, 5, 6, 6, Yes], [6, 6, 5, 5, Yes], [6, 7, 4, 4, Yes], [6, 8, 3, 3, Yes], [6, 9, 2, 2, Yes], [6, 10, 1, 1, Yes], [6, 11, 0, 0, Yes], [6, 12, 16, 16, Yes], [6, 13, 15, 15, Yes], [6, 14, 14, 14, Yes], [6, 15, 13, 13, Yes], [6, 16, 12, 12, Yes], [7, 0, 10, 10, Yes], [7, 1, 9, 9, Yes], [7, 2, 8, 8, Yes], [7, 3, 7, 7, Yes], [7, 4, 6, 6, Yes], [7, 5, 5, 5, Yes], [7, 6, 4, 4, Yes], [7, 7, 3, 3, Yes], [7, 8, 2, 2, Yes], [7, 9, 1, 1, Yes], [7, 10, 0, 0, Yes], [7, 11, 16, 16, Yes], [7, 12, 15, 15, Yes], [7, 13, 14, 14, Yes], [7, 14, 13, 13, Yes], [7, 15, 12, 12, Yes], [7, 16, 11, 11, Yes], [8, 0, 9, 9, Yes], [8, 1, 8, 8, Yes], [8, 2, 7, 7, Yes], [8, 3, 6, 6, Yes], [8, 4, 5, 5, Yes], [8, 5, 4, 4, Yes], [8, 6, 3, 3, Yes], [8, 7, 2, 2, Yes], [8, 8, 1, 1, Yes], [8, 9, 0, 0, Yes], [8, 10, 16, 16, Yes], [8, 11, 15, 15, Yes], [8, 12, 14, 14, Yes], [8, 13, 13, 13, Yes], [8, 14, 12, 12, Yes], [8, 15, 11, 11, Yes], [8, 16, 10, 10, Yes], [9, 0, 8, 8, Yes], [9, 1, 7, 7, Yes], [9, 2, 6, 6, Yes], [9, 3, 5, 5, Yes], [9, 4, 4, 4, Yes], [9, 5, 3, 3, Yes], [9, 6, 2, 2, Yes], [9, 7, 1, 1, Yes], [9, 8, 0, 0, Yes], [9, 9, 16, 16, Yes], [9, 10, 15, 15, Yes], [9, 11, 14, 14, Yes], [9, 12, 13, 13, Yes], [9, 13, 12, 12, Yes], [9, 14, 11, 11, Yes], [9, 15, 10, 10, Yes], [9, 16, 9, 9, Yes], [10, 0, 7, 7, Yes], [10, 1, 6, 6, Yes], [10, 2, 5, 5, Yes], [10, 3, 4, 4, Yes], [10, 4, 3, 3, Yes], [10, 5, 2, 2, Yes], [10, 6, 1, 1, Yes], [10, 7, 0, 0, Yes], [10, 8, 16, 16, Yes], [10, 9, 15, 15, Yes], [10, 10, 14, 14, Yes], [10, 11, 13, 13, Yes], [10, 12, 12, 12, Yes], [10, 13, 11, 11, Yes], [10, 14, 10, 10, Yes], [10, 15, 9, 9, Yes], [10, 16, 8, 8, Yes], [11, 0, 6, 6, Yes], [11, 1, 5, 5, Yes], [11, 2, 4, 4, Yes], [11, 3, 3, 3, Yes], [11, 4, 2, 2, Yes], [11, 5, 1, 1, Yes], [11, 6, 0, 0, Yes], [11, 7, 16, 16, Yes], [11, 8, 15, 15, Yes], [11, 9, 14, 14, Yes], [11, 10, 13, 13, Yes], [11, 11, 12, 12, Yes], [11, 12, 11, 11, Yes], [11, 13, 10, 10, Yes], [11, 14, 9, 9, Yes], [11, 15, 8, 8, Yes], [11, 16, 7, 7, Yes], [12, 0, 5, 5, Yes], [12, 1, 4, 4, Yes], [12, 2, 3, 3, Yes], [12, 3, 2, 2, Yes], [12, 4, 1, 1, Yes], [12, 5, 0, 0, Yes], [12, 6, 16, 16, Yes], [12, 7, 15, 15, Yes], [12, 8, 14, 14, Yes], [12, 9, 13, 13, Yes], [12, 10, 12, 12, Yes], [12, 11, 11, 11, Yes], [12, 12, 10, 10, Yes], [12, 13, 9, 9, Yes], [12, 14, 8, 8, Yes], [12, 15, 7, 7, Yes], [12, 16, 6, 6, Yes], [13, 0, 4, 4, Yes], [13, 1, 3, 3, Yes], [13, 2, 2, 2, Yes], [13, 3, 1, 1, Yes], [13, 4, 0, 0, Yes], [13, 5, 16, 16, Yes], [13, 6, 15, 15, Yes], [13, 7, 14, 14, Yes], [13, 8, 13, 13, Yes], [13, 9, 12, 12, Yes], [13, 10, 11, 11, Yes], [13, 11, 10, 10, Yes], [13, 12, 9, 9, Yes], [13, 13, 8, 8, Yes], [13, 14, 7, 7, Yes], [13, 15, 6, 6, Yes], [13, 16, 5, 5, Yes], [14, 0, 3, 3, Yes], [14, 1, 2, 2, Yes], [14, 2, 1, 1, Yes], [14, 3, 0, 0, Yes], [14, 4, 16, 16, Yes], [14, 5, 15, 15, Yes], [14, 6, 14, 14, Yes], [14, 7, 13, 13, Yes], [14, 8, 12, 12, Yes], [14, 9, 11, 11, Yes], [14, 10, 10, 10, Yes], [14, 11, 9, 9, Yes], [14, 12, 8, 8, Yes], [14, 13, 7, 7, Yes], [14, 14, 6, 6, Yes], [14, 15, 5, 5, Yes], [14, 16, 4, 4, Yes], [15, 0, 2, 2, Yes], [15, 1, 1, 1, Yes], [15, 2, 0, 0, Yes], [15, 3, 16, 16, Yes], [15, 4, 15, 15, Yes], [15, 5, 14, 14, Yes], [15, 6, 13, 13, Yes], [15, 7, 12, 12, Yes], [15, 8, 11, 11, Yes], [15, 9, 10, 10, Yes], [15, 10, 9, 9, Yes], [15, 11, 8, 8, Yes], [15, 12, 7, 7, Yes], [15, 13, 6, 6, Yes], [15, 14, 5, 5, Yes], [15, 15, 4, 4, Yes], [15, 16, 3, 3, Yes], [16, 0, 1, 1, Yes], [16, 1, 0, 0, Yes], [16, 2, 16, 16, Yes], [16, 3, 15, 15, Yes], [16, 4, 14, 14, Yes], [16, 5, 13, 13, Yes], [16, 6, 12, 12, Yes], [16, 7, 11, 11, Yes], [16, 8, 10, 10, Yes], [16, 9, 9, 9, Yes], [16, 10, 8, 8, Yes], [16, 11, 7, 7, Yes], [16, 12, 6, 6, Yes], [16, 13, 5, 5, Yes], [16, 14, 4, 4, Yes], [16, 15, 3, 3, Yes], [16, 16, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (17, 17, [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 16, 16, Yes], [0, 2, 15, 15, Yes], [0, 3, 14, 14, Yes], [0, 4, 13, 13, Yes], [0, 5, 12, 12, Yes], [0, 6, 11, 11, Yes], [0, 7, 10, 10, Yes], [0, 8, 9, 9, Yes], [0, 9, 8, 8, Yes], [0, 10, 7, 7, Yes], [0, 11, 6, 6, Yes], [0, 12, 5, 5, Yes], [0, 13, 4, 4, Yes], [0, 14, 3, 3, Yes], [0, 15, 2, 2, Yes], [0, 16, 1, 1, Yes], [1, 0, 16, 16, Yes], [1, 1, 15, 15, Yes], [1, 2, 14, 14, Yes], [1, 3, 13, 13, Yes], [1, 4, 12, 12, Yes], [1, 5, 11, 11, Yes], [1, 6, 10, 10, Yes], [1, 7, 9, 9, Yes], [1, 8, 8, 8, Yes], [1, 9, 7, 7, Yes], [1, 10, 6, 6, Yes], [1, 11, 5, 5, Yes], [1, 12, 4, 4, Yes], [1, 13, 3, 3, Yes], [1, 14, 2, 2, Yes], [1, 15, 1, 1, Yes], [1, 16, 0, 0, Yes], [2, 0, 15, 15, Yes], [2, 1, 14, 14, Yes], [2, 2, 13, 13, Yes], [2, 3, 12, 12, Yes], [2, 4, 11, 11, Yes], [2, 5, 10, 10, Yes], [2, 6, 9, 9, Yes], [2, 7, 8, 8, Yes], [2, 8, 7, 7, Yes], [2, 9, 6, 6, Yes], [2, 10, 5, 5, Yes], [2, 11, 4, 4, Yes], [2, 12, 3, 3, Yes], [2, 13, 2, 2, Yes], [2, 14, 1, 1, Yes], [2, 15, 0, 0, Yes], [2, 16, 16, 16, Yes], [3, 0, 14, 14, Yes], [3, 1, 13, 13, Yes], [3, 2, 12, 12, Yes], [3, 3, 11, 11, Yes], [3, 4, 10, 10, Yes], [3, 5, 9, 9, Yes], [3, 6, 8, 8, Yes], [3, 7, 7, 7, Yes], [3, 8, 6, 6, Yes], [3, 9, 5, 5, Yes], [3, 10, 4, 4, Yes], [3, 11, 3, 3, Yes], [3, 12, 2, 2, Yes], [3, 13, 1, 1, Yes], [3, 14, 0, 0, Yes], [3, 15, 16, 16, Yes], [3, 16, 15, 15, Yes], [4, 0, 13, 13, Yes], [4, 1, 12, 12, Yes], [4, 2, 11, 11, Yes], [4, 3, 10, 10, Yes], [4, 4, 9, 9, Yes], [4, 5, 8, 8, Yes], [4, 6, 7, 7, Yes], [4, 7, 6, 6, Yes], [4, 8, 5, 5, Yes], [4, 9, 4, 4, Yes], [4, 10, 3, 3, Yes], [4, 11, 2, 2, Yes], [4, 12, 1, 1, Yes], [4, 13, 0, 0, Yes], [4, 14, 16, 16, Yes], [4, 15, 15, 15, Yes], [4, 16, 14, 14, Yes], [5, 0, 12, 12, Yes], [5, 1, 11, 11, Yes], [5, 2, 10, 10, Yes], [5, 3, 9, 9, Yes], [5, 4, 8, 8, Yes], [5, 5, 7, 7, Yes], [5, 6, 6, 6, Yes], [5, 7, 5, 5, Yes], [5, 8, 4, 4, Yes], [5, 9, 3, 3, Yes], [5, 10, 2, 2, Yes], [5, 11, 1, 1, Yes], [5, 12, 0, 0, Yes], [5, 13, 16, 16, Yes], [5, 14, 15, 15, Yes], [5, 15, 14, 14, Yes], [5, 16, 13, 13, Yes], [6, 0, 11, 11, Yes], [6, 1, 10, 10, Yes], [6, 2, 9, 9, Yes], [6, 3, 8, 8, Yes], [6, 4, 7, 7, Yes], [6, 5, 6, 6, Yes], [6, 6, 5, 5, Yes], [6, 7, 4, 4, Yes], [6, 8, 3, 3, Yes], [6, 9, 2, 2, Yes], [6, 10, 1, 1, Yes], [6, 11, 0, 0, Yes], [6, 12, 16, 16, Yes], [6, 13, 15, 15, Yes], [6, 14, 14, 14, Yes], [6, 15, 13, 13, Yes], [6, 16, 12, 12, Yes], [7, 0, 10, 10, Yes], [7, 1, 9, 9, Yes], [7, 2, 8, 8, Yes], [7, 3, 7, 7, Yes], [7, 4, 6, 6, Yes], [7, 5, 5, 5, Yes], [7, 6, 4, 4, Yes], [7, 7, 3, 3, Yes], [7, 8, 2, 2, Yes], [7, 9, 1, 1, Yes], [7, 10, 0, 0, Yes], [7, 11, 16, 16, Yes], [7, 12, 15, 15, Yes], [7, 13, 14, 14, Yes], [7, 14, 13, 13, Yes], [7, 15, 12, 12, Yes], [7, 16, 11, 11, Yes], [8, 0, 9, 9, Yes], [8, 1, 8, 8, Yes], [8, 2, 7, 7, Yes], [8, 3, 6, 6, Yes], [8, 4, 5, 5, Yes], [8, 5, 4, 4, Yes], [8, 6, 3, 3, Yes], [8, 7, 2, 2, Yes], [8, 8, 1, 1, Yes], [8, 9, 0, 0, Yes], [8, 10, 16, 16, Yes], [8, 11, 15, 15, Yes], [8, 12, 14, 14, Yes], [8, 13, 13, 13, Yes], [8, 14, 12, 12, Yes], [8, 15, 11, 11, Yes], [8, 16, 10, 10, Yes], [9, 0, 8, 8, Yes], [9, 1, 7, 7, Yes], [9, 2, 6, 6, Yes], [9, 3, 5, 5, Yes], [9, 4, 4, 4, Yes], [9, 5, 3, 3, Yes], [9, 6, 2, 2, Yes], [9, 7, 1, 1, Yes], [9, 8, 0, 0, Yes], [9, 9, 16, 16, Yes], [9, 10, 15, 15, Yes], [9, 11, 14, 14, Yes], [9, 12, 13, 13, Yes], [9, 13, 12, 12, Yes], [9, 14, 11, 11, Yes], [9, 15, 10, 10, Yes], [9, 16, 9, 9, Yes], [10, 0, 7, 7, Yes], [10, 1, 6, 6, Yes], [10, 2, 5, 5, Yes], [10, 3, 4, 4, Yes], [10, 4, 3, 3, Yes], [10, 5, 2, 2, Yes], [10, 6, 1, 1, Yes], [10, 7, 0, 0, Yes], [10, 8, 16, 16, Yes], [10, 9, 15, 15, Yes], [10, 10, 14, 14, Yes], [10, 11, 13, 13, Yes], [10, 12, 12, 12, Yes], [10, 13, 11, 11, Yes], [10, 14, 10, 10, Yes], [10, 15, 9, 9, Yes], [10, 16, 8, 8, Yes], [11, 0, 6, 6, Yes], [11, 1, 5, 5, Yes], [11, 2, 4, 4, Yes], [11, 3, 3, 3, Yes], [11, 4, 2, 2, Yes], [11, 5, 1, 1, Yes], [11, 6, 0, 0, Yes], [11, 7, 16, 16, Yes], [11, 8, 15, 15, Yes], [11, 9, 14, 14, Yes], [11, 10, 13, 13, Yes], [11, 11, 12, 12, Yes], [11, 12, 11, 11, Yes], [11, 13, 10, 10, Yes], [11, 14, 9, 9, Yes], [11, 15, 8, 8, Yes], [11, 16, 7, 7, Yes], [12, 0, 5, 5, Yes], [12, 1, 4, 4, Yes], [12, 2, 3, 3, Yes], [12, 3, 2, 2, Yes], [12, 4, 1, 1, Yes], [12, 5, 0, 0, Yes], [12, 6, 16, 16, Yes], [12, 7, 15, 15, Yes], [12, 8, 14, 14, Yes], [12, 9, 13, 13, Yes], [12, 10, 12, 12, Yes], [12, 11, 11, 11, Yes], [12, 12, 10, 10, Yes], [12, 13, 9, 9, Yes], [12, 14, 8, 8, Yes], [12, 15, 7, 7, Yes], [12, 16, 6, 6, Yes], [13, 0, 4, 4, Yes], [13, 1, 3, 3, Yes], [13, 2, 2, 2, Yes], [13, 3, 1, 1, Yes], [13, 4, 0, 0, Yes], [13, 5, 16, 16, Yes], [13, 6, 15, 15, Yes], [13, 7, 14, 14, Yes], [13, 8, 13, 13, Yes], [13, 9, 12, 12, Yes], [13, 10, 11, 11, Yes], [13, 11, 10, 10, Yes], [13, 12, 9, 9, Yes], [13, 13, 8, 8, Yes], [13, 14, 7, 7, Yes], [13, 15, 6, 6, Yes], [13, 16, 5, 5, Yes], [14, 0, 3, 3, Yes], [14, 1, 2, 2, Yes], [14, 2, 1, 1, Yes], [14, 3, 0, 0, Yes], [14, 4, 16, 16, Yes], [14, 5, 15, 15, Yes], [14, 6, 14, 14, Yes], [14, 7, 13, 13, Yes], [14, 8, 12, 12, Yes], [14, 9, 11, 11, Yes], [14, 10, 10, 10, Yes], [14, 11, 9, 9, Yes], [14, 12, 8, 8, Yes], [14, 13, 7, 7, Yes], [14, 14, 6, 6, Yes], [14, 15, 5, 5, Yes], [14, 16, 4, 4, Yes], [15, 0, 2, 2, Yes], [15, 1, 1, 1, Yes], [15, 2, 0, 0, Yes], [15, 3, 16, 16, Yes], [15, 4, 15, 15, Yes], [15, 5, 14, 14, Yes], [15, 6, 13, 13, Yes], [15, 7, 12, 12, Yes], [15, 8, 11, 11, Yes], [15, 9, 10, 10, Yes], [15, 10, 9, 9, Yes], [15, 11, 8, 8, Yes], [15, 12, 7, 7, Yes], [15, 13, 6, 6, Yes], [15, 14, 5, 5, Yes], [15, 15, 4, 4, Yes], [15, 16, 3, 3, Yes], [16, 0, 1, 1, Yes], [16, 1, 0, 0, Yes], [16, 2, 16, 16, Yes], [16, 3, 15, 15, Yes], [16, 4, 14, 14, Yes], [16, 5, 13, 13, Yes], [16, 6, 12, 12, Yes], [16, 7, 11, 11, Yes], [16, 8, 10, 10, Yes], [16, 9, 9, 9, Yes], [16, 10, 8, 8, Yes], [16, 11, 7, 7, Yes], [16, 12, 6, 6, Yes], [16, 13, 5, 5, Yes], [16, 14, 4, 4, Yes], [16, 15, 3, 3, Yes], [16, 16, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0016","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7572e777425c840e1eb3ebe8cba0018d2f01067c8563ce5242316091879a5e62","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141016,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], target identity=0, and total mapping=[0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(18, 18, [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 17, 17, Yes], [0, 2, 16, 16, Yes], [0, 3, 15, 15, Yes], [0, 4, 14, 14, Yes], [0, 5, 13, 13, Yes], [0, 6, 12, 12, Yes], [0, 7, 11, 11, Yes], [0, 8, 10, 10, Yes], [0, 9, 9, 9, Yes], [0, 10, 8, 8, Yes], [0, 11, 7, 7, Yes], [0, 12, 6, 6, Yes], [0, 13, 5, 5, Yes], [0, 14, 4, 4, Yes], [0, 15, 3, 3, Yes], [0, 16, 2, 2, Yes], [0, 17, 1, 1, Yes], [1, 0, 17, 17, Yes], [1, 1, 16, 16, Yes], [1, 2, 15, 15, Yes], [1, 3, 14, 14, Yes], [1, 4, 13, 13, Yes], [1, 5, 12, 12, Yes], [1, 6, 11, 11, Yes], [1, 7, 10, 10, Yes], [1, 8, 9, 9, Yes], [1, 9, 8, 8, Yes], [1, 10, 7, 7, Yes], [1, 11, 6, 6, Yes], [1, 12, 5, 5, Yes], [1, 13, 4, 4, Yes], [1, 14, 3, 3, Yes], [1, 15, 2, 2, Yes], [1, 16, 1, 1, Yes], [1, 17, 0, 0, Yes], [2, 0, 16, 16, Yes], [2, 1, 15, 15, Yes], [2, 2, 14, 14, Yes], [2, 3, 13, 13, Yes], [2, 4, 12, 12, Yes], [2, 5, 11, 11, Yes], [2, 6, 10, 10, Yes], [2, 7, 9, 9, Yes], [2, 8, 8, 8, Yes], [2, 9, 7, 7, Yes], [2, 10, 6, 6, Yes], [2, 11, 5, 5, Yes], [2, 12, 4, 4, Yes], [2, 13, 3, 3, Yes], [2, 14, 2, 2, Yes], [2, 15, 1, 1, Yes], [2, 16, 0, 0, Yes], [2, 17, 17, 17, Yes], [3, 0, 15, 15, Yes], [3, 1, 14, 14, Yes], [3, 2, 13, 13, Yes], [3, 3, 12, 12, Yes], [3, 4, 11, 11, Yes], [3, 5, 10, 10, Yes], [3, 6, 9, 9, Yes], [3, 7, 8, 8, Yes], [3, 8, 7, 7, Yes], [3, 9, 6, 6, Yes], [3, 10, 5, 5, Yes], [3, 11, 4, 4, Yes], [3, 12, 3, 3, Yes], [3, 13, 2, 2, Yes], [3, 14, 1, 1, Yes], [3, 15, 0, 0, Yes], [3, 16, 17, 17, Yes], [3, 17, 16, 16, Yes], [4, 0, 14, 14, Yes], [4, 1, 13, 13, Yes], [4, 2, 12, 12, Yes], [4, 3, 11, 11, Yes], [4, 4, 10, 10, Yes], [4, 5, 9, 9, Yes], [4, 6, 8, 8, Yes], [4, 7, 7, 7, Yes], [4, 8, 6, 6, Yes], [4, 9, 5, 5, Yes], [4, 10, 4, 4, Yes], [4, 11, 3, 3, Yes], [4, 12, 2, 2, Yes], [4, 13, 1, 1, Yes], [4, 14, 0, 0, Yes], [4, 15, 17, 17, Yes], [4, 16, 16, 16, Yes], [4, 17, 15, 15, Yes], [5, 0, 13, 13, Yes], [5, 1, 12, 12, Yes], [5, 2, 11, 11, Yes], [5, 3, 10, 10, Yes], [5, 4, 9, 9, Yes], [5, 5, 8, 8, Yes], [5, 6, 7, 7, Yes], [5, 7, 6, 6, Yes], [5, 8, 5, 5, Yes], [5, 9, 4, 4, Yes], [5, 10, 3, 3, Yes], [5, 11, 2, 2, Yes], [5, 12, 1, 1, Yes], [5, 13, 0, 0, Yes], [5, 14, 17, 17, Yes], [5, 15, 16, 16, Yes], [5, 16, 15, 15, Yes], [5, 17, 14, 14, Yes], [6, 0, 12, 12, Yes], [6, 1, 11, 11, Yes], [6, 2, 10, 10, Yes], [6, 3, 9, 9, Yes], [6, 4, 8, 8, Yes], [6, 5, 7, 7, Yes], [6, 6, 6, 6, Yes], [6, 7, 5, 5, Yes], [6, 8, 4, 4, Yes], [6, 9, 3, 3, Yes], [6, 10, 2, 2, Yes], [6, 11, 1, 1, Yes], [6, 12, 0, 0, Yes], [6, 13, 17, 17, Yes], [6, 14, 16, 16, Yes], [6, 15, 15, 15, Yes], [6, 16, 14, 14, Yes], [6, 17, 13, 13, Yes], [7, 0, 11, 11, Yes], [7, 1, 10, 10, Yes], [7, 2, 9, 9, Yes], [7, 3, 8, 8, Yes], [7, 4, 7, 7, Yes], [7, 5, 6, 6, Yes], [7, 6, 5, 5, Yes], [7, 7, 4, 4, Yes], [7, 8, 3, 3, Yes], [7, 9, 2, 2, Yes], [7, 10, 1, 1, Yes], [7, 11, 0, 0, Yes], [7, 12, 17, 17, Yes], [7, 13, 16, 16, Yes], [7, 14, 15, 15, Yes], [7, 15, 14, 14, Yes], [7, 16, 13, 13, Yes], [7, 17, 12, 12, Yes], [8, 0, 10, 10, Yes], [8, 1, 9, 9, Yes], [8, 2, 8, 8, Yes], [8, 3, 7, 7, Yes], [8, 4, 6, 6, Yes], [8, 5, 5, 5, Yes], [8, 6, 4, 4, Yes], [8, 7, 3, 3, Yes], [8, 8, 2, 2, Yes], [8, 9, 1, 1, Yes], [8, 10, 0, 0, Yes], [8, 11, 17, 17, Yes], [8, 12, 16, 16, Yes], [8, 13, 15, 15, Yes], [8, 14, 14, 14, Yes], [8, 15, 13, 13, Yes], [8, 16, 12, 12, Yes], [8, 17, 11, 11, Yes], [9, 0, 9, 9, Yes], [9, 1, 8, 8, Yes], [9, 2, 7, 7, Yes], [9, 3, 6, 6, Yes], [9, 4, 5, 5, Yes], [9, 5, 4, 4, Yes], [9, 6, 3, 3, Yes], [9, 7, 2, 2, Yes], [9, 8, 1, 1, Yes], [9, 9, 0, 0, Yes], [9, 10, 17, 17, Yes], [9, 11, 16, 16, Yes], [9, 12, 15, 15, Yes], [9, 13, 14, 14, Yes], [9, 14, 13, 13, Yes], [9, 15, 12, 12, Yes], [9, 16, 11, 11, Yes], [9, 17, 10, 10, Yes], [10, 0, 8, 8, Yes], [10, 1, 7, 7, Yes], [10, 2, 6, 6, Yes], [10, 3, 5, 5, Yes], [10, 4, 4, 4, Yes], [10, 5, 3, 3, Yes], [10, 6, 2, 2, Yes], [10, 7, 1, 1, Yes], [10, 8, 0, 0, Yes], [10, 9, 17, 17, Yes], [10, 10, 16, 16, Yes], [10, 11, 15, 15, Yes], [10, 12, 14, 14, Yes], [10, 13, 13, 13, Yes], [10, 14, 12, 12, Yes], [10, 15, 11, 11, Yes], [10, 16, 10, 10, Yes], [10, 17, 9, 9, Yes], [11, 0, 7, 7, Yes], [11, 1, 6, 6, Yes], [11, 2, 5, 5, Yes], [11, 3, 4, 4, Yes], [11, 4, 3, 3, Yes], [11, 5, 2, 2, Yes], [11, 6, 1, 1, Yes], [11, 7, 0, 0, Yes], [11, 8, 17, 17, Yes], [11, 9, 16, 16, Yes], [11, 10, 15, 15, Yes], [11, 11, 14, 14, Yes], [11, 12, 13, 13, Yes], [11, 13, 12, 12, Yes], [11, 14, 11, 11, Yes], [11, 15, 10, 10, Yes], [11, 16, 9, 9, Yes], [11, 17, 8, 8, Yes], [12, 0, 6, 6, Yes], [12, 1, 5, 5, Yes], [12, 2, 4, 4, Yes], [12, 3, 3, 3, Yes], [12, 4, 2, 2, Yes], [12, 5, 1, 1, Yes], [12, 6, 0, 0, Yes], [12, 7, 17, 17, Yes], [12, 8, 16, 16, Yes], [12, 9, 15, 15, Yes], [12, 10, 14, 14, Yes], [12, 11, 13, 13, Yes], [12, 12, 12, 12, Yes], [12, 13, 11, 11, Yes], [12, 14, 10, 10, Yes], [12, 15, 9, 9, Yes], [12, 16, 8, 8, Yes], [12, 17, 7, 7, Yes], [13, 0, 5, 5, Yes], [13, 1, 4, 4, Yes], [13, 2, 3, 3, Yes], [13, 3, 2, 2, Yes], [13, 4, 1, 1, Yes], [13, 5, 0, 0, Yes], [13, 6, 17, 17, Yes], [13, 7, 16, 16, Yes], [13, 8, 15, 15, Yes], [13, 9, 14, 14, Yes], [13, 10, 13, 13, Yes], [13, 11, 12, 12, Yes], [13, 12, 11, 11, Yes], [13, 13, 10, 10, Yes], [13, 14, 9, 9, Yes], [13, 15, 8, 8, Yes], [13, 16, 7, 7, Yes], [13, 17, 6, 6, Yes], [14, 0, 4, 4, Yes], [14, 1, 3, 3, Yes], [14, 2, 2, 2, Yes], [14, 3, 1, 1, Yes], [14, 4, 0, 0, Yes], [14, 5, 17, 17, Yes], [14, 6, 16, 16, Yes], [14, 7, 15, 15, Yes], [14, 8, 14, 14, Yes], [14, 9, 13, 13, Yes], [14, 10, 12, 12, Yes], [14, 11, 11, 11, Yes], [14, 12, 10, 10, Yes], [14, 13, 9, 9, Yes], [14, 14, 8, 8, Yes], [14, 15, 7, 7, Yes], [14, 16, 6, 6, Yes], [14, 17, 5, 5, Yes], [15, 0, 3, 3, Yes], [15, 1, 2, 2, Yes], [15, 2, 1, 1, Yes], [15, 3, 0, 0, Yes], [15, 4, 17, 17, Yes], [15, 5, 16, 16, Yes], [15, 6, 15, 15, Yes], [15, 7, 14, 14, Yes], [15, 8, 13, 13, Yes], [15, 9, 12, 12, Yes], [15, 10, 11, 11, Yes], [15, 11, 10, 10, Yes], [15, 12, 9, 9, Yes], [15, 13, 8, 8, Yes], [15, 14, 7, 7, Yes], [15, 15, 6, 6, Yes], [15, 16, 5, 5, Yes], [15, 17, 4, 4, Yes], [16, 0, 2, 2, Yes], [16, 1, 1, 1, Yes], [16, 2, 0, 0, Yes], [16, 3, 17, 17, Yes], [16, 4, 16, 16, Yes], [16, 5, 15, 15, Yes], [16, 6, 14, 14, Yes], [16, 7, 13, 13, Yes], [16, 8, 12, 12, Yes], [16, 9, 11, 11, Yes], [16, 10, 10, 10, Yes], [16, 11, 9, 9, Yes], [16, 12, 8, 8, Yes], [16, 13, 7, 7, Yes], [16, 14, 6, 6, Yes], [16, 15, 5, 5, Yes], [16, 16, 4, 4, Yes], [16, 17, 3, 3, Yes], [17, 0, 1, 1, Yes], [17, 1, 0, 0, Yes], [17, 2, 17, 17, Yes], [17, 3, 16, 16, Yes], [17, 4, 15, 15, Yes], [17, 5, 14, 14, Yes], [17, 6, 13, 13, Yes], [17, 7, 12, 12, Yes], [17, 8, 11, 11, Yes], [17, 9, 10, 10, Yes], [17, 10, 9, 9, Yes], [17, 11, 8, 8, Yes], [17, 12, 7, 7, Yes], [17, 13, 6, 6, Yes], [17, 14, 5, 5, Yes], [17, 15, 4, 4, Yes], [17, 16, 3, 3, Yes], [17, 17, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (18, 18, [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 17, 17, Yes], [0, 2, 16, 16, Yes], [0, 3, 15, 15, Yes], [0, 4, 14, 14, Yes], [0, 5, 13, 13, Yes], [0, 6, 12, 12, Yes], [0, 7, 11, 11, Yes], [0, 8, 10, 10, Yes], [0, 9, 9, 9, Yes], [0, 10, 8, 8, Yes], [0, 11, 7, 7, Yes], [0, 12, 6, 6, Yes], [0, 13, 5, 5, Yes], [0, 14, 4, 4, Yes], [0, 15, 3, 3, Yes], [0, 16, 2, 2, Yes], [0, 17, 1, 1, Yes], [1, 0, 17, 17, Yes], [1, 1, 16, 16, Yes], [1, 2, 15, 15, Yes], [1, 3, 14, 14, Yes], [1, 4, 13, 13, Yes], [1, 5, 12, 12, Yes], [1, 6, 11, 11, Yes], [1, 7, 10, 10, Yes], [1, 8, 9, 9, Yes], [1, 9, 8, 8, Yes], [1, 10, 7, 7, Yes], [1, 11, 6, 6, Yes], [1, 12, 5, 5, Yes], [1, 13, 4, 4, Yes], [1, 14, 3, 3, Yes], [1, 15, 2, 2, Yes], [1, 16, 1, 1, Yes], [1, 17, 0, 0, Yes], [2, 0, 16, 16, Yes], [2, 1, 15, 15, Yes], [2, 2, 14, 14, Yes], [2, 3, 13, 13, Yes], [2, 4, 12, 12, Yes], [2, 5, 11, 11, Yes], [2, 6, 10, 10, Yes], [2, 7, 9, 9, Yes], [2, 8, 8, 8, Yes], [2, 9, 7, 7, Yes], [2, 10, 6, 6, Yes], [2, 11, 5, 5, Yes], [2, 12, 4, 4, Yes], [2, 13, 3, 3, Yes], [2, 14, 2, 2, Yes], [2, 15, 1, 1, Yes], [2, 16, 0, 0, Yes], [2, 17, 17, 17, Yes], [3, 0, 15, 15, Yes], [3, 1, 14, 14, Yes], [3, 2, 13, 13, Yes], [3, 3, 12, 12, Yes], [3, 4, 11, 11, Yes], [3, 5, 10, 10, Yes], [3, 6, 9, 9, Yes], [3, 7, 8, 8, Yes], [3, 8, 7, 7, Yes], [3, 9, 6, 6, Yes], [3, 10, 5, 5, Yes], [3, 11, 4, 4, Yes], [3, 12, 3, 3, Yes], [3, 13, 2, 2, Yes], [3, 14, 1, 1, Yes], [3, 15, 0, 0, Yes], [3, 16, 17, 17, Yes], [3, 17, 16, 16, Yes], [4, 0, 14, 14, Yes], [4, 1, 13, 13, Yes], [4, 2, 12, 12, Yes], [4, 3, 11, 11, Yes], [4, 4, 10, 10, Yes], [4, 5, 9, 9, Yes], [4, 6, 8, 8, Yes], [4, 7, 7, 7, Yes], [4, 8, 6, 6, Yes], [4, 9, 5, 5, Yes], [4, 10, 4, 4, Yes], [4, 11, 3, 3, Yes], [4, 12, 2, 2, Yes], [4, 13, 1, 1, Yes], [4, 14, 0, 0, Yes], [4, 15, 17, 17, Yes], [4, 16, 16, 16, Yes], [4, 17, 15, 15, Yes], [5, 0, 13, 13, Yes], [5, 1, 12, 12, Yes], [5, 2, 11, 11, Yes], [5, 3, 10, 10, Yes], [5, 4, 9, 9, Yes], [5, 5, 8, 8, Yes], [5, 6, 7, 7, Yes], [5, 7, 6, 6, Yes], [5, 8, 5, 5, Yes], [5, 9, 4, 4, Yes], [5, 10, 3, 3, Yes], [5, 11, 2, 2, Yes], [5, 12, 1, 1, Yes], [5, 13, 0, 0, Yes], [5, 14, 17, 17, Yes], [5, 15, 16, 16, Yes], [5, 16, 15, 15, Yes], [5, 17, 14, 14, Yes], [6, 0, 12, 12, Yes], [6, 1, 11, 11, Yes], [6, 2, 10, 10, Yes], [6, 3, 9, 9, Yes], [6, 4, 8, 8, Yes], [6, 5, 7, 7, Yes], [6, 6, 6, 6, Yes], [6, 7, 5, 5, Yes], [6, 8, 4, 4, Yes], [6, 9, 3, 3, Yes], [6, 10, 2, 2, Yes], [6, 11, 1, 1, Yes], [6, 12, 0, 0, Yes], [6, 13, 17, 17, Yes], [6, 14, 16, 16, Yes], [6, 15, 15, 15, Yes], [6, 16, 14, 14, Yes], [6, 17, 13, 13, Yes], [7, 0, 11, 11, Yes], [7, 1, 10, 10, Yes], [7, 2, 9, 9, Yes], [7, 3, 8, 8, Yes], [7, 4, 7, 7, Yes], [7, 5, 6, 6, Yes], [7, 6, 5, 5, Yes], [7, 7, 4, 4, Yes], [7, 8, 3, 3, Yes], [7, 9, 2, 2, Yes], [7, 10, 1, 1, Yes], [7, 11, 0, 0, Yes], [7, 12, 17, 17, Yes], [7, 13, 16, 16, Yes], [7, 14, 15, 15, Yes], [7, 15, 14, 14, Yes], [7, 16, 13, 13, Yes], [7, 17, 12, 12, Yes], [8, 0, 10, 10, Yes], [8, 1, 9, 9, Yes], [8, 2, 8, 8, Yes], [8, 3, 7, 7, Yes], [8, 4, 6, 6, Yes], [8, 5, 5, 5, Yes], [8, 6, 4, 4, Yes], [8, 7, 3, 3, Yes], [8, 8, 2, 2, Yes], [8, 9, 1, 1, Yes], [8, 10, 0, 0, Yes], [8, 11, 17, 17, Yes], [8, 12, 16, 16, Yes], [8, 13, 15, 15, Yes], [8, 14, 14, 14, Yes], [8, 15, 13, 13, Yes], [8, 16, 12, 12, Yes], [8, 17, 11, 11, Yes], [9, 0, 9, 9, Yes], [9, 1, 8, 8, Yes], [9, 2, 7, 7, Yes], [9, 3, 6, 6, Yes], [9, 4, 5, 5, Yes], [9, 5, 4, 4, Yes], [9, 6, 3, 3, Yes], [9, 7, 2, 2, Yes], [9, 8, 1, 1, Yes], [9, 9, 0, 0, Yes], [9, 10, 17, 17, Yes], [9, 11, 16, 16, Yes], [9, 12, 15, 15, Yes], [9, 13, 14, 14, Yes], [9, 14, 13, 13, Yes], [9, 15, 12, 12, Yes], [9, 16, 11, 11, Yes], [9, 17, 10, 10, Yes], [10, 0, 8, 8, Yes], [10, 1, 7, 7, Yes], [10, 2, 6, 6, Yes], [10, 3, 5, 5, Yes], [10, 4, 4, 4, Yes], [10, 5, 3, 3, Yes], [10, 6, 2, 2, Yes], [10, 7, 1, 1, Yes], [10, 8, 0, 0, Yes], [10, 9, 17, 17, Yes], [10, 10, 16, 16, Yes], [10, 11, 15, 15, Yes], [10, 12, 14, 14, Yes], [10, 13, 13, 13, Yes], [10, 14, 12, 12, Yes], [10, 15, 11, 11, Yes], [10, 16, 10, 10, Yes], [10, 17, 9, 9, Yes], [11, 0, 7, 7, Yes], [11, 1, 6, 6, Yes], [11, 2, 5, 5, Yes], [11, 3, 4, 4, Yes], [11, 4, 3, 3, Yes], [11, 5, 2, 2, Yes], [11, 6, 1, 1, Yes], [11, 7, 0, 0, Yes], [11, 8, 17, 17, Yes], [11, 9, 16, 16, Yes], [11, 10, 15, 15, Yes], [11, 11, 14, 14, Yes], [11, 12, 13, 13, Yes], [11, 13, 12, 12, Yes], [11, 14, 11, 11, Yes], [11, 15, 10, 10, Yes], [11, 16, 9, 9, Yes], [11, 17, 8, 8, Yes], [12, 0, 6, 6, Yes], [12, 1, 5, 5, Yes], [12, 2, 4, 4, Yes], [12, 3, 3, 3, Yes], [12, 4, 2, 2, Yes], [12, 5, 1, 1, Yes], [12, 6, 0, 0, Yes], [12, 7, 17, 17, Yes], [12, 8, 16, 16, Yes], [12, 9, 15, 15, Yes], [12, 10, 14, 14, Yes], [12, 11, 13, 13, Yes], [12, 12, 12, 12, Yes], [12, 13, 11, 11, Yes], [12, 14, 10, 10, Yes], [12, 15, 9, 9, Yes], [12, 16, 8, 8, Yes], [12, 17, 7, 7, Yes], [13, 0, 5, 5, Yes], [13, 1, 4, 4, Yes], [13, 2, 3, 3, Yes], [13, 3, 2, 2, Yes], [13, 4, 1, 1, Yes], [13, 5, 0, 0, Yes], [13, 6, 17, 17, Yes], [13, 7, 16, 16, Yes], [13, 8, 15, 15, Yes], [13, 9, 14, 14, Yes], [13, 10, 13, 13, Yes], [13, 11, 12, 12, Yes], [13, 12, 11, 11, Yes], [13, 13, 10, 10, Yes], [13, 14, 9, 9, Yes], [13, 15, 8, 8, Yes], [13, 16, 7, 7, Yes], [13, 17, 6, 6, Yes], [14, 0, 4, 4, Yes], [14, 1, 3, 3, Yes], [14, 2, 2, 2, Yes], [14, 3, 1, 1, Yes], [14, 4, 0, 0, Yes], [14, 5, 17, 17, Yes], [14, 6, 16, 16, Yes], [14, 7, 15, 15, Yes], [14, 8, 14, 14, Yes], [14, 9, 13, 13, Yes], [14, 10, 12, 12, Yes], [14, 11, 11, 11, Yes], [14, 12, 10, 10, Yes], [14, 13, 9, 9, Yes], [14, 14, 8, 8, Yes], [14, 15, 7, 7, Yes], [14, 16, 6, 6, Yes], [14, 17, 5, 5, Yes], [15, 0, 3, 3, Yes], [15, 1, 2, 2, Yes], [15, 2, 1, 1, Yes], [15, 3, 0, 0, Yes], [15, 4, 17, 17, Yes], [15, 5, 16, 16, Yes], [15, 6, 15, 15, Yes], [15, 7, 14, 14, Yes], [15, 8, 13, 13, Yes], [15, 9, 12, 12, Yes], [15, 10, 11, 11, Yes], [15, 11, 10, 10, Yes], [15, 12, 9, 9, Yes], [15, 13, 8, 8, Yes], [15, 14, 7, 7, Yes], [15, 15, 6, 6, Yes], [15, 16, 5, 5, Yes], [15, 17, 4, 4, Yes], [16, 0, 2, 2, Yes], [16, 1, 1, 1, Yes], [16, 2, 0, 0, Yes], [16, 3, 17, 17, Yes], [16, 4, 16, 16, Yes], [16, 5, 15, 15, Yes], [16, 6, 14, 14, Yes], [16, 7, 13, 13, Yes], [16, 8, 12, 12, Yes], [16, 9, 11, 11, Yes], [16, 10, 10, 10, Yes], [16, 11, 9, 9, Yes], [16, 12, 8, 8, Yes], [16, 13, 7, 7, Yes], [16, 14, 6, 6, Yes], [16, 15, 5, 5, Yes], [16, 16, 4, 4, Yes], [16, 17, 3, 3, Yes], [17, 0, 1, 1, Yes], [17, 1, 0, 0, Yes], [17, 2, 17, 17, Yes], [17, 3, 16, 16, Yes], [17, 4, 15, 15, Yes], [17, 5, 14, 14, Yes], [17, 6, 13, 13, Yes], [17, 7, 12, 12, Yes], [17, 8, 11, 11, Yes], [17, 9, 10, 10, Yes], [17, 10, 9, 9, Yes], [17, 11, 8, 8, Yes], [17, 12, 7, 7, Yes], [17, 13, 6, 6, Yes], [17, 14, 5, 5, Yes], [17, 15, 4, 4, Yes], [17, 16, 3, 3, Yes], [17, 17, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0017","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a364f50f1af72f70117675f4260cd66d4b9736e88e816fb7ef7f65692248b3e0","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141017,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], target identity=0, and total mapping=[0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(19, 19, [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 18, 18, Yes], [0, 2, 17, 17, Yes], [0, 3, 16, 16, Yes], [0, 4, 15, 15, Yes], [0, 5, 14, 14, Yes], [0, 6, 13, 13, Yes], [0, 7, 12, 12, Yes], [0, 8, 11, 11, Yes], [0, 9, 10, 10, Yes], [0, 10, 9, 9, Yes], [0, 11, 8, 8, Yes], [0, 12, 7, 7, Yes], [0, 13, 6, 6, Yes], [0, 14, 5, 5, Yes], [0, 15, 4, 4, Yes], [0, 16, 3, 3, Yes], [0, 17, 2, 2, Yes], [0, 18, 1, 1, Yes], [1, 0, 18, 18, Yes], [1, 1, 17, 17, Yes], [1, 2, 16, 16, Yes], [1, 3, 15, 15, Yes], [1, 4, 14, 14, Yes], [1, 5, 13, 13, Yes], [1, 6, 12, 12, Yes], [1, 7, 11, 11, Yes], [1, 8, 10, 10, Yes], [1, 9, 9, 9, Yes], [1, 10, 8, 8, Yes], [1, 11, 7, 7, Yes], [1, 12, 6, 6, Yes], [1, 13, 5, 5, Yes], [1, 14, 4, 4, Yes], [1, 15, 3, 3, Yes], [1, 16, 2, 2, Yes], [1, 17, 1, 1, Yes], [1, 18, 0, 0, Yes], [2, 0, 17, 17, Yes], [2, 1, 16, 16, Yes], [2, 2, 15, 15, Yes], [2, 3, 14, 14, Yes], [2, 4, 13, 13, Yes], [2, 5, 12, 12, Yes], [2, 6, 11, 11, Yes], [2, 7, 10, 10, Yes], [2, 8, 9, 9, Yes], [2, 9, 8, 8, Yes], [2, 10, 7, 7, Yes], [2, 11, 6, 6, Yes], [2, 12, 5, 5, Yes], [2, 13, 4, 4, Yes], [2, 14, 3, 3, Yes], [2, 15, 2, 2, Yes], [2, 16, 1, 1, Yes], [2, 17, 0, 0, Yes], [2, 18, 18, 18, Yes], [3, 0, 16, 16, Yes], [3, 1, 15, 15, Yes], [3, 2, 14, 14, Yes], [3, 3, 13, 13, Yes], [3, 4, 12, 12, Yes], [3, 5, 11, 11, Yes], [3, 6, 10, 10, Yes], [3, 7, 9, 9, Yes], [3, 8, 8, 8, Yes], [3, 9, 7, 7, Yes], [3, 10, 6, 6, Yes], [3, 11, 5, 5, Yes], [3, 12, 4, 4, Yes], [3, 13, 3, 3, Yes], [3, 14, 2, 2, Yes], [3, 15, 1, 1, Yes], [3, 16, 0, 0, Yes], [3, 17, 18, 18, Yes], [3, 18, 17, 17, Yes], [4, 0, 15, 15, Yes], [4, 1, 14, 14, Yes], [4, 2, 13, 13, Yes], [4, 3, 12, 12, Yes], [4, 4, 11, 11, Yes], [4, 5, 10, 10, Yes], [4, 6, 9, 9, Yes], [4, 7, 8, 8, Yes], [4, 8, 7, 7, Yes], [4, 9, 6, 6, Yes], [4, 10, 5, 5, Yes], [4, 11, 4, 4, Yes], [4, 12, 3, 3, Yes], [4, 13, 2, 2, Yes], [4, 14, 1, 1, Yes], [4, 15, 0, 0, Yes], [4, 16, 18, 18, Yes], [4, 17, 17, 17, Yes], [4, 18, 16, 16, Yes], [5, 0, 14, 14, Yes], [5, 1, 13, 13, Yes], [5, 2, 12, 12, Yes], [5, 3, 11, 11, Yes], [5, 4, 10, 10, Yes], [5, 5, 9, 9, Yes], [5, 6, 8, 8, Yes], [5, 7, 7, 7, Yes], [5, 8, 6, 6, Yes], [5, 9, 5, 5, Yes], [5, 10, 4, 4, Yes], [5, 11, 3, 3, Yes], [5, 12, 2, 2, Yes], [5, 13, 1, 1, Yes], [5, 14, 0, 0, Yes], [5, 15, 18, 18, Yes], [5, 16, 17, 17, Yes], [5, 17, 16, 16, Yes], [5, 18, 15, 15, Yes], [6, 0, 13, 13, Yes], [6, 1, 12, 12, Yes], [6, 2, 11, 11, Yes], [6, 3, 10, 10, Yes], [6, 4, 9, 9, Yes], [6, 5, 8, 8, Yes], [6, 6, 7, 7, Yes], [6, 7, 6, 6, Yes], [6, 8, 5, 5, Yes], [6, 9, 4, 4, Yes], [6, 10, 3, 3, Yes], [6, 11, 2, 2, Yes], [6, 12, 1, 1, Yes], [6, 13, 0, 0, Yes], [6, 14, 18, 18, Yes], [6, 15, 17, 17, Yes], [6, 16, 16, 16, Yes], [6, 17, 15, 15, Yes], [6, 18, 14, 14, Yes], [7, 0, 12, 12, Yes], [7, 1, 11, 11, Yes], [7, 2, 10, 10, Yes], [7, 3, 9, 9, Yes], [7, 4, 8, 8, Yes], [7, 5, 7, 7, Yes], [7, 6, 6, 6, Yes], [7, 7, 5, 5, Yes], [7, 8, 4, 4, Yes], [7, 9, 3, 3, Yes], [7, 10, 2, 2, Yes], [7, 11, 1, 1, Yes], [7, 12, 0, 0, Yes], [7, 13, 18, 18, Yes], [7, 14, 17, 17, Yes], [7, 15, 16, 16, Yes], [7, 16, 15, 15, Yes], [7, 17, 14, 14, Yes], [7, 18, 13, 13, Yes], [8, 0, 11, 11, Yes], [8, 1, 10, 10, Yes], [8, 2, 9, 9, Yes], [8, 3, 8, 8, Yes], [8, 4, 7, 7, Yes], [8, 5, 6, 6, Yes], [8, 6, 5, 5, Yes], [8, 7, 4, 4, Yes], [8, 8, 3, 3, Yes], [8, 9, 2, 2, Yes], [8, 10, 1, 1, Yes], [8, 11, 0, 0, Yes], [8, 12, 18, 18, Yes], [8, 13, 17, 17, Yes], [8, 14, 16, 16, Yes], [8, 15, 15, 15, Yes], [8, 16, 14, 14, Yes], [8, 17, 13, 13, Yes], [8, 18, 12, 12, Yes], [9, 0, 10, 10, Yes], [9, 1, 9, 9, Yes], [9, 2, 8, 8, Yes], [9, 3, 7, 7, Yes], [9, 4, 6, 6, Yes], [9, 5, 5, 5, Yes], [9, 6, 4, 4, Yes], [9, 7, 3, 3, Yes], [9, 8, 2, 2, Yes], [9, 9, 1, 1, Yes], [9, 10, 0, 0, Yes], [9, 11, 18, 18, Yes], [9, 12, 17, 17, Yes], [9, 13, 16, 16, Yes], [9, 14, 15, 15, Yes], [9, 15, 14, 14, Yes], [9, 16, 13, 13, Yes], [9, 17, 12, 12, Yes], [9, 18, 11, 11, Yes], [10, 0, 9, 9, Yes], [10, 1, 8, 8, Yes], [10, 2, 7, 7, Yes], [10, 3, 6, 6, Yes], [10, 4, 5, 5, Yes], [10, 5, 4, 4, Yes], [10, 6, 3, 3, Yes], [10, 7, 2, 2, Yes], [10, 8, 1, 1, Yes], [10, 9, 0, 0, Yes], [10, 10, 18, 18, Yes], [10, 11, 17, 17, Yes], [10, 12, 16, 16, Yes], [10, 13, 15, 15, Yes], [10, 14, 14, 14, Yes], [10, 15, 13, 13, Yes], [10, 16, 12, 12, Yes], [10, 17, 11, 11, Yes], [10, 18, 10, 10, Yes], [11, 0, 8, 8, Yes], [11, 1, 7, 7, Yes], [11, 2, 6, 6, Yes], [11, 3, 5, 5, Yes], [11, 4, 4, 4, Yes], [11, 5, 3, 3, Yes], [11, 6, 2, 2, Yes], [11, 7, 1, 1, Yes], [11, 8, 0, 0, Yes], [11, 9, 18, 18, Yes], [11, 10, 17, 17, Yes], [11, 11, 16, 16, Yes], [11, 12, 15, 15, Yes], [11, 13, 14, 14, Yes], [11, 14, 13, 13, Yes], [11, 15, 12, 12, Yes], [11, 16, 11, 11, Yes], [11, 17, 10, 10, Yes], [11, 18, 9, 9, Yes], [12, 0, 7, 7, Yes], [12, 1, 6, 6, Yes], [12, 2, 5, 5, Yes], [12, 3, 4, 4, Yes], [12, 4, 3, 3, Yes], [12, 5, 2, 2, Yes], [12, 6, 1, 1, Yes], [12, 7, 0, 0, Yes], [12, 8, 18, 18, Yes], [12, 9, 17, 17, Yes], [12, 10, 16, 16, Yes], [12, 11, 15, 15, Yes], [12, 12, 14, 14, Yes], [12, 13, 13, 13, Yes], [12, 14, 12, 12, Yes], [12, 15, 11, 11, Yes], [12, 16, 10, 10, Yes], [12, 17, 9, 9, Yes], [12, 18, 8, 8, Yes], [13, 0, 6, 6, Yes], [13, 1, 5, 5, Yes], [13, 2, 4, 4, Yes], [13, 3, 3, 3, Yes], [13, 4, 2, 2, Yes], [13, 5, 1, 1, Yes], [13, 6, 0, 0, Yes], [13, 7, 18, 18, Yes], [13, 8, 17, 17, Yes], [13, 9, 16, 16, Yes], [13, 10, 15, 15, Yes], [13, 11, 14, 14, Yes], [13, 12, 13, 13, Yes], [13, 13, 12, 12, Yes], [13, 14, 11, 11, Yes], [13, 15, 10, 10, Yes], [13, 16, 9, 9, Yes], [13, 17, 8, 8, Yes], [13, 18, 7, 7, Yes], [14, 0, 5, 5, Yes], [14, 1, 4, 4, Yes], [14, 2, 3, 3, Yes], [14, 3, 2, 2, Yes], [14, 4, 1, 1, Yes], [14, 5, 0, 0, Yes], [14, 6, 18, 18, Yes], [14, 7, 17, 17, Yes], [14, 8, 16, 16, Yes], [14, 9, 15, 15, Yes], [14, 10, 14, 14, Yes], [14, 11, 13, 13, Yes], [14, 12, 12, 12, Yes], [14, 13, 11, 11, Yes], [14, 14, 10, 10, Yes], [14, 15, 9, 9, Yes], [14, 16, 8, 8, Yes], [14, 17, 7, 7, Yes], [14, 18, 6, 6, Yes], [15, 0, 4, 4, Yes], [15, 1, 3, 3, Yes], [15, 2, 2, 2, Yes], [15, 3, 1, 1, Yes], [15, 4, 0, 0, Yes], [15, 5, 18, 18, Yes], [15, 6, 17, 17, Yes], [15, 7, 16, 16, Yes], [15, 8, 15, 15, Yes], [15, 9, 14, 14, Yes], [15, 10, 13, 13, Yes], [15, 11, 12, 12, Yes], [15, 12, 11, 11, Yes], [15, 13, 10, 10, Yes], [15, 14, 9, 9, Yes], [15, 15, 8, 8, Yes], [15, 16, 7, 7, Yes], [15, 17, 6, 6, Yes], [15, 18, 5, 5, Yes], [16, 0, 3, 3, Yes], [16, 1, 2, 2, Yes], [16, 2, 1, 1, Yes], [16, 3, 0, 0, Yes], [16, 4, 18, 18, Yes], [16, 5, 17, 17, Yes], [16, 6, 16, 16, Yes], [16, 7, 15, 15, Yes], [16, 8, 14, 14, Yes], [16, 9, 13, 13, Yes], [16, 10, 12, 12, Yes], [16, 11, 11, 11, Yes], [16, 12, 10, 10, Yes], [16, 13, 9, 9, Yes], [16, 14, 8, 8, Yes], [16, 15, 7, 7, Yes], [16, 16, 6, 6, Yes], [16, 17, 5, 5, Yes], [16, 18, 4, 4, Yes], [17, 0, 2, 2, Yes], [17, 1, 1, 1, Yes], [17, 2, 0, 0, Yes], [17, 3, 18, 18, Yes], [17, 4, 17, 17, Yes], [17, 5, 16, 16, Yes], [17, 6, 15, 15, Yes], [17, 7, 14, 14, Yes], [17, 8, 13, 13, Yes], [17, 9, 12, 12, Yes], [17, 10, 11, 11, Yes], [17, 11, 10, 10, Yes], [17, 12, 9, 9, Yes], [17, 13, 8, 8, Yes], [17, 14, 7, 7, Yes], [17, 15, 6, 6, Yes], [17, 16, 5, 5, Yes], [17, 17, 4, 4, Yes], [17, 18, 3, 3, Yes], [18, 0, 1, 1, Yes], [18, 1, 0, 0, Yes], [18, 2, 18, 18, Yes], [18, 3, 17, 17, Yes], [18, 4, 16, 16, Yes], [18, 5, 15, 15, Yes], [18, 6, 14, 14, Yes], [18, 7, 13, 13, Yes], [18, 8, 12, 12, Yes], [18, 9, 11, 11, Yes], [18, 10, 10, 10, Yes], [18, 11, 9, 9, Yes], [18, 12, 8, 8, Yes], [18, 13, 7, 7, Yes], [18, 14, 6, 6, Yes], [18, 15, 5, 5, Yes], [18, 16, 4, 4, Yes], [18, 17, 3, 3, Yes], [18, 18, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (19, 19, [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 18, 18, Yes], [0, 2, 17, 17, Yes], [0, 3, 16, 16, Yes], [0, 4, 15, 15, Yes], [0, 5, 14, 14, Yes], [0, 6, 13, 13, Yes], [0, 7, 12, 12, Yes], [0, 8, 11, 11, Yes], [0, 9, 10, 10, Yes], [0, 10, 9, 9, Yes], [0, 11, 8, 8, Yes], [0, 12, 7, 7, Yes], [0, 13, 6, 6, Yes], [0, 14, 5, 5, Yes], [0, 15, 4, 4, Yes], [0, 16, 3, 3, Yes], [0, 17, 2, 2, Yes], [0, 18, 1, 1, Yes], [1, 0, 18, 18, Yes], [1, 1, 17, 17, Yes], [1, 2, 16, 16, Yes], [1, 3, 15, 15, Yes], [1, 4, 14, 14, Yes], [1, 5, 13, 13, Yes], [1, 6, 12, 12, Yes], [1, 7, 11, 11, Yes], [1, 8, 10, 10, Yes], [1, 9, 9, 9, Yes], [1, 10, 8, 8, Yes], [1, 11, 7, 7, Yes], [1, 12, 6, 6, Yes], [1, 13, 5, 5, Yes], [1, 14, 4, 4, Yes], [1, 15, 3, 3, Yes], [1, 16, 2, 2, Yes], [1, 17, 1, 1, Yes], [1, 18, 0, 0, Yes], [2, 0, 17, 17, Yes], [2, 1, 16, 16, Yes], [2, 2, 15, 15, Yes], [2, 3, 14, 14, Yes], [2, 4, 13, 13, Yes], [2, 5, 12, 12, Yes], [2, 6, 11, 11, Yes], [2, 7, 10, 10, Yes], [2, 8, 9, 9, Yes], [2, 9, 8, 8, Yes], [2, 10, 7, 7, Yes], [2, 11, 6, 6, Yes], [2, 12, 5, 5, Yes], [2, 13, 4, 4, Yes], [2, 14, 3, 3, Yes], [2, 15, 2, 2, Yes], [2, 16, 1, 1, Yes], [2, 17, 0, 0, Yes], [2, 18, 18, 18, Yes], [3, 0, 16, 16, Yes], [3, 1, 15, 15, Yes], [3, 2, 14, 14, Yes], [3, 3, 13, 13, Yes], [3, 4, 12, 12, Yes], [3, 5, 11, 11, Yes], [3, 6, 10, 10, Yes], [3, 7, 9, 9, Yes], [3, 8, 8, 8, Yes], [3, 9, 7, 7, Yes], [3, 10, 6, 6, Yes], [3, 11, 5, 5, Yes], [3, 12, 4, 4, Yes], [3, 13, 3, 3, Yes], [3, 14, 2, 2, Yes], [3, 15, 1, 1, Yes], [3, 16, 0, 0, Yes], [3, 17, 18, 18, Yes], [3, 18, 17, 17, Yes], [4, 0, 15, 15, Yes], [4, 1, 14, 14, Yes], [4, 2, 13, 13, Yes], [4, 3, 12, 12, Yes], [4, 4, 11, 11, Yes], [4, 5, 10, 10, Yes], [4, 6, 9, 9, Yes], [4, 7, 8, 8, Yes], [4, 8, 7, 7, Yes], [4, 9, 6, 6, Yes], [4, 10, 5, 5, Yes], [4, 11, 4, 4, Yes], [4, 12, 3, 3, Yes], [4, 13, 2, 2, Yes], [4, 14, 1, 1, Yes], [4, 15, 0, 0, Yes], [4, 16, 18, 18, Yes], [4, 17, 17, 17, Yes], [4, 18, 16, 16, Yes], [5, 0, 14, 14, Yes], [5, 1, 13, 13, Yes], [5, 2, 12, 12, Yes], [5, 3, 11, 11, Yes], [5, 4, 10, 10, Yes], [5, 5, 9, 9, Yes], [5, 6, 8, 8, Yes], [5, 7, 7, 7, Yes], [5, 8, 6, 6, Yes], [5, 9, 5, 5, Yes], [5, 10, 4, 4, Yes], [5, 11, 3, 3, Yes], [5, 12, 2, 2, Yes], [5, 13, 1, 1, Yes], [5, 14, 0, 0, Yes], [5, 15, 18, 18, Yes], [5, 16, 17, 17, Yes], [5, 17, 16, 16, Yes], [5, 18, 15, 15, Yes], [6, 0, 13, 13, Yes], [6, 1, 12, 12, Yes], [6, 2, 11, 11, Yes], [6, 3, 10, 10, Yes], [6, 4, 9, 9, Yes], [6, 5, 8, 8, Yes], [6, 6, 7, 7, Yes], [6, 7, 6, 6, Yes], [6, 8, 5, 5, Yes], [6, 9, 4, 4, Yes], [6, 10, 3, 3, Yes], [6, 11, 2, 2, Yes], [6, 12, 1, 1, Yes], [6, 13, 0, 0, Yes], [6, 14, 18, 18, Yes], [6, 15, 17, 17, Yes], [6, 16, 16, 16, Yes], [6, 17, 15, 15, Yes], [6, 18, 14, 14, Yes], [7, 0, 12, 12, Yes], [7, 1, 11, 11, Yes], [7, 2, 10, 10, Yes], [7, 3, 9, 9, Yes], [7, 4, 8, 8, Yes], [7, 5, 7, 7, Yes], [7, 6, 6, 6, Yes], [7, 7, 5, 5, Yes], [7, 8, 4, 4, Yes], [7, 9, 3, 3, Yes], [7, 10, 2, 2, Yes], [7, 11, 1, 1, Yes], [7, 12, 0, 0, Yes], [7, 13, 18, 18, Yes], [7, 14, 17, 17, Yes], [7, 15, 16, 16, Yes], [7, 16, 15, 15, Yes], [7, 17, 14, 14, Yes], [7, 18, 13, 13, Yes], [8, 0, 11, 11, Yes], [8, 1, 10, 10, Yes], [8, 2, 9, 9, Yes], [8, 3, 8, 8, Yes], [8, 4, 7, 7, Yes], [8, 5, 6, 6, Yes], [8, 6, 5, 5, Yes], [8, 7, 4, 4, Yes], [8, 8, 3, 3, Yes], [8, 9, 2, 2, Yes], [8, 10, 1, 1, Yes], [8, 11, 0, 0, Yes], [8, 12, 18, 18, Yes], [8, 13, 17, 17, Yes], [8, 14, 16, 16, Yes], [8, 15, 15, 15, Yes], [8, 16, 14, 14, Yes], [8, 17, 13, 13, Yes], [8, 18, 12, 12, Yes], [9, 0, 10, 10, Yes], [9, 1, 9, 9, Yes], [9, 2, 8, 8, Yes], [9, 3, 7, 7, Yes], [9, 4, 6, 6, Yes], [9, 5, 5, 5, Yes], [9, 6, 4, 4, Yes], [9, 7, 3, 3, Yes], [9, 8, 2, 2, Yes], [9, 9, 1, 1, Yes], [9, 10, 0, 0, Yes], [9, 11, 18, 18, Yes], [9, 12, 17, 17, Yes], [9, 13, 16, 16, Yes], [9, 14, 15, 15, Yes], [9, 15, 14, 14, Yes], [9, 16, 13, 13, Yes], [9, 17, 12, 12, Yes], [9, 18, 11, 11, Yes], [10, 0, 9, 9, Yes], [10, 1, 8, 8, Yes], [10, 2, 7, 7, Yes], [10, 3, 6, 6, Yes], [10, 4, 5, 5, Yes], [10, 5, 4, 4, Yes], [10, 6, 3, 3, Yes], [10, 7, 2, 2, Yes], [10, 8, 1, 1, Yes], [10, 9, 0, 0, Yes], [10, 10, 18, 18, Yes], [10, 11, 17, 17, Yes], [10, 12, 16, 16, Yes], [10, 13, 15, 15, Yes], [10, 14, 14, 14, Yes], [10, 15, 13, 13, Yes], [10, 16, 12, 12, Yes], [10, 17, 11, 11, Yes], [10, 18, 10, 10, Yes], [11, 0, 8, 8, Yes], [11, 1, 7, 7, Yes], [11, 2, 6, 6, Yes], [11, 3, 5, 5, Yes], [11, 4, 4, 4, Yes], [11, 5, 3, 3, Yes], [11, 6, 2, 2, Yes], [11, 7, 1, 1, Yes], [11, 8, 0, 0, Yes], [11, 9, 18, 18, Yes], [11, 10, 17, 17, Yes], [11, 11, 16, 16, Yes], [11, 12, 15, 15, Yes], [11, 13, 14, 14, Yes], [11, 14, 13, 13, Yes], [11, 15, 12, 12, Yes], [11, 16, 11, 11, Yes], [11, 17, 10, 10, Yes], [11, 18, 9, 9, Yes], [12, 0, 7, 7, Yes], [12, 1, 6, 6, Yes], [12, 2, 5, 5, Yes], [12, 3, 4, 4, Yes], [12, 4, 3, 3, Yes], [12, 5, 2, 2, Yes], [12, 6, 1, 1, Yes], [12, 7, 0, 0, Yes], [12, 8, 18, 18, Yes], [12, 9, 17, 17, Yes], [12, 10, 16, 16, Yes], [12, 11, 15, 15, Yes], [12, 12, 14, 14, Yes], [12, 13, 13, 13, Yes], [12, 14, 12, 12, Yes], [12, 15, 11, 11, Yes], [12, 16, 10, 10, Yes], [12, 17, 9, 9, Yes], [12, 18, 8, 8, Yes], [13, 0, 6, 6, Yes], [13, 1, 5, 5, Yes], [13, 2, 4, 4, Yes], [13, 3, 3, 3, Yes], [13, 4, 2, 2, Yes], [13, 5, 1, 1, Yes], [13, 6, 0, 0, Yes], [13, 7, 18, 18, Yes], [13, 8, 17, 17, Yes], [13, 9, 16, 16, Yes], [13, 10, 15, 15, Yes], [13, 11, 14, 14, Yes], [13, 12, 13, 13, Yes], [13, 13, 12, 12, Yes], [13, 14, 11, 11, Yes], [13, 15, 10, 10, Yes], [13, 16, 9, 9, Yes], [13, 17, 8, 8, Yes], [13, 18, 7, 7, Yes], [14, 0, 5, 5, Yes], [14, 1, 4, 4, Yes], [14, 2, 3, 3, Yes], [14, 3, 2, 2, Yes], [14, 4, 1, 1, Yes], [14, 5, 0, 0, Yes], [14, 6, 18, 18, Yes], [14, 7, 17, 17, Yes], [14, 8, 16, 16, Yes], [14, 9, 15, 15, Yes], [14, 10, 14, 14, Yes], [14, 11, 13, 13, Yes], [14, 12, 12, 12, Yes], [14, 13, 11, 11, Yes], [14, 14, 10, 10, Yes], [14, 15, 9, 9, Yes], [14, 16, 8, 8, Yes], [14, 17, 7, 7, Yes], [14, 18, 6, 6, Yes], [15, 0, 4, 4, Yes], [15, 1, 3, 3, Yes], [15, 2, 2, 2, Yes], [15, 3, 1, 1, Yes], [15, 4, 0, 0, Yes], [15, 5, 18, 18, Yes], [15, 6, 17, 17, Yes], [15, 7, 16, 16, Yes], [15, 8, 15, 15, Yes], [15, 9, 14, 14, Yes], [15, 10, 13, 13, Yes], [15, 11, 12, 12, Yes], [15, 12, 11, 11, Yes], [15, 13, 10, 10, Yes], [15, 14, 9, 9, Yes], [15, 15, 8, 8, Yes], [15, 16, 7, 7, Yes], [15, 17, 6, 6, Yes], [15, 18, 5, 5, Yes], [16, 0, 3, 3, Yes], [16, 1, 2, 2, Yes], [16, 2, 1, 1, Yes], [16, 3, 0, 0, Yes], [16, 4, 18, 18, Yes], [16, 5, 17, 17, Yes], [16, 6, 16, 16, Yes], [16, 7, 15, 15, Yes], [16, 8, 14, 14, Yes], [16, 9, 13, 13, Yes], [16, 10, 12, 12, Yes], [16, 11, 11, 11, Yes], [16, 12, 10, 10, Yes], [16, 13, 9, 9, Yes], [16, 14, 8, 8, Yes], [16, 15, 7, 7, Yes], [16, 16, 6, 6, Yes], [16, 17, 5, 5, Yes], [16, 18, 4, 4, Yes], [17, 0, 2, 2, Yes], [17, 1, 1, 1, Yes], [17, 2, 0, 0, Yes], [17, 3, 18, 18, Yes], [17, 4, 17, 17, Yes], [17, 5, 16, 16, Yes], [17, 6, 15, 15, Yes], [17, 7, 14, 14, Yes], [17, 8, 13, 13, Yes], [17, 9, 12, 12, Yes], [17, 10, 11, 11, Yes], [17, 11, 10, 10, Yes], [17, 12, 9, 9, Yes], [17, 13, 8, 8, Yes], [17, 14, 7, 7, Yes], [17, 15, 6, 6, Yes], [17, 16, 5, 5, Yes], [17, 17, 4, 4, Yes], [17, 18, 3, 3, Yes], [18, 0, 1, 1, Yes], [18, 1, 0, 0, Yes], [18, 2, 18, 18, Yes], [18, 3, 17, 17, Yes], [18, 4, 16, 16, Yes], [18, 5, 15, 15, Yes], [18, 6, 14, 14, Yes], [18, 7, 13, 13, Yes], [18, 8, 12, 12, Yes], [18, 9, 11, 11, Yes], [18, 10, 10, 10, Yes], [18, 11, 9, 9, Yes], [18, 12, 8, 8, Yes], [18, 13, 7, 7, Yes], [18, 14, 6, 6, Yes], [18, 15, 5, 5, Yes], [18, 16, 4, 4, Yes], [18, 17, 3, 3, Yes], [18, 18, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0018","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a40a4f7d2f8cc601c71f7e11f8e8f628dece48c7a498550a78991a4cc9b7f3a","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141018,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], target identity=0, and total mapping=[0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(20, 20, [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 19, 19, Yes], [0, 2, 18, 18, Yes], [0, 3, 17, 17, Yes], [0, 4, 16, 16, Yes], [0, 5, 15, 15, Yes], [0, 6, 14, 14, Yes], [0, 7, 13, 13, Yes], [0, 8, 12, 12, Yes], [0, 9, 11, 11, Yes], [0, 10, 10, 10, Yes], [0, 11, 9, 9, Yes], [0, 12, 8, 8, Yes], [0, 13, 7, 7, Yes], [0, 14, 6, 6, Yes], [0, 15, 5, 5, Yes], [0, 16, 4, 4, Yes], [0, 17, 3, 3, Yes], [0, 18, 2, 2, Yes], [0, 19, 1, 1, Yes], [1, 0, 19, 19, Yes], [1, 1, 18, 18, Yes], [1, 2, 17, 17, Yes], [1, 3, 16, 16, Yes], [1, 4, 15, 15, Yes], [1, 5, 14, 14, Yes], [1, 6, 13, 13, Yes], [1, 7, 12, 12, Yes], [1, 8, 11, 11, Yes], [1, 9, 10, 10, Yes], [1, 10, 9, 9, Yes], [1, 11, 8, 8, Yes], [1, 12, 7, 7, Yes], [1, 13, 6, 6, Yes], [1, 14, 5, 5, Yes], [1, 15, 4, 4, Yes], [1, 16, 3, 3, Yes], [1, 17, 2, 2, Yes], [1, 18, 1, 1, Yes], [1, 19, 0, 0, Yes], [2, 0, 18, 18, Yes], [2, 1, 17, 17, Yes], [2, 2, 16, 16, Yes], [2, 3, 15, 15, Yes], [2, 4, 14, 14, Yes], [2, 5, 13, 13, Yes], [2, 6, 12, 12, Yes], [2, 7, 11, 11, Yes], [2, 8, 10, 10, Yes], [2, 9, 9, 9, Yes], [2, 10, 8, 8, Yes], [2, 11, 7, 7, Yes], [2, 12, 6, 6, Yes], [2, 13, 5, 5, Yes], [2, 14, 4, 4, Yes], [2, 15, 3, 3, Yes], [2, 16, 2, 2, Yes], [2, 17, 1, 1, Yes], [2, 18, 0, 0, Yes], [2, 19, 19, 19, Yes], [3, 0, 17, 17, Yes], [3, 1, 16, 16, Yes], [3, 2, 15, 15, Yes], [3, 3, 14, 14, Yes], [3, 4, 13, 13, Yes], [3, 5, 12, 12, Yes], [3, 6, 11, 11, Yes], [3, 7, 10, 10, Yes], [3, 8, 9, 9, Yes], [3, 9, 8, 8, Yes], [3, 10, 7, 7, Yes], [3, 11, 6, 6, Yes], [3, 12, 5, 5, Yes], [3, 13, 4, 4, Yes], [3, 14, 3, 3, Yes], [3, 15, 2, 2, Yes], [3, 16, 1, 1, Yes], [3, 17, 0, 0, Yes], [3, 18, 19, 19, Yes], [3, 19, 18, 18, Yes], [4, 0, 16, 16, Yes], [4, 1, 15, 15, Yes], [4, 2, 14, 14, Yes], [4, 3, 13, 13, Yes], [4, 4, 12, 12, Yes], [4, 5, 11, 11, Yes], [4, 6, 10, 10, Yes], [4, 7, 9, 9, Yes], [4, 8, 8, 8, Yes], [4, 9, 7, 7, Yes], [4, 10, 6, 6, Yes], [4, 11, 5, 5, Yes], [4, 12, 4, 4, Yes], [4, 13, 3, 3, Yes], [4, 14, 2, 2, Yes], [4, 15, 1, 1, Yes], [4, 16, 0, 0, Yes], [4, 17, 19, 19, Yes], [4, 18, 18, 18, Yes], [4, 19, 17, 17, Yes], [5, 0, 15, 15, Yes], [5, 1, 14, 14, Yes], [5, 2, 13, 13, Yes], [5, 3, 12, 12, Yes], [5, 4, 11, 11, Yes], [5, 5, 10, 10, Yes], [5, 6, 9, 9, Yes], [5, 7, 8, 8, Yes], [5, 8, 7, 7, Yes], [5, 9, 6, 6, Yes], [5, 10, 5, 5, Yes], [5, 11, 4, 4, Yes], [5, 12, 3, 3, Yes], [5, 13, 2, 2, Yes], [5, 14, 1, 1, Yes], [5, 15, 0, 0, Yes], [5, 16, 19, 19, Yes], [5, 17, 18, 18, Yes], [5, 18, 17, 17, Yes], [5, 19, 16, 16, Yes], [6, 0, 14, 14, Yes], [6, 1, 13, 13, Yes], [6, 2, 12, 12, Yes], [6, 3, 11, 11, Yes], [6, 4, 10, 10, Yes], [6, 5, 9, 9, Yes], [6, 6, 8, 8, Yes], [6, 7, 7, 7, Yes], [6, 8, 6, 6, Yes], [6, 9, 5, 5, Yes], [6, 10, 4, 4, Yes], [6, 11, 3, 3, Yes], [6, 12, 2, 2, Yes], [6, 13, 1, 1, Yes], [6, 14, 0, 0, Yes], [6, 15, 19, 19, Yes], [6, 16, 18, 18, Yes], [6, 17, 17, 17, Yes], [6, 18, 16, 16, Yes], [6, 19, 15, 15, Yes], [7, 0, 13, 13, Yes], [7, 1, 12, 12, Yes], [7, 2, 11, 11, Yes], [7, 3, 10, 10, Yes], [7, 4, 9, 9, Yes], [7, 5, 8, 8, Yes], [7, 6, 7, 7, Yes], [7, 7, 6, 6, Yes], [7, 8, 5, 5, Yes], [7, 9, 4, 4, Yes], [7, 10, 3, 3, Yes], [7, 11, 2, 2, Yes], [7, 12, 1, 1, Yes], [7, 13, 0, 0, Yes], [7, 14, 19, 19, Yes], [7, 15, 18, 18, Yes], [7, 16, 17, 17, Yes], [7, 17, 16, 16, Yes], [7, 18, 15, 15, Yes], [7, 19, 14, 14, Yes], [8, 0, 12, 12, Yes], [8, 1, 11, 11, Yes], [8, 2, 10, 10, Yes], [8, 3, 9, 9, Yes], [8, 4, 8, 8, Yes], [8, 5, 7, 7, Yes], [8, 6, 6, 6, Yes], [8, 7, 5, 5, Yes], [8, 8, 4, 4, Yes], [8, 9, 3, 3, Yes], [8, 10, 2, 2, Yes], [8, 11, 1, 1, Yes], [8, 12, 0, 0, Yes], [8, 13, 19, 19, Yes], [8, 14, 18, 18, Yes], [8, 15, 17, 17, Yes], [8, 16, 16, 16, Yes], [8, 17, 15, 15, Yes], [8, 18, 14, 14, Yes], [8, 19, 13, 13, Yes], [9, 0, 11, 11, Yes], [9, 1, 10, 10, Yes], [9, 2, 9, 9, Yes], [9, 3, 8, 8, Yes], [9, 4, 7, 7, Yes], [9, 5, 6, 6, Yes], [9, 6, 5, 5, Yes], [9, 7, 4, 4, Yes], [9, 8, 3, 3, Yes], [9, 9, 2, 2, Yes], [9, 10, 1, 1, Yes], [9, 11, 0, 0, Yes], [9, 12, 19, 19, Yes], [9, 13, 18, 18, Yes], [9, 14, 17, 17, Yes], [9, 15, 16, 16, Yes], [9, 16, 15, 15, Yes], [9, 17, 14, 14, Yes], [9, 18, 13, 13, Yes], [9, 19, 12, 12, Yes], [10, 0, 10, 10, Yes], [10, 1, 9, 9, Yes], [10, 2, 8, 8, Yes], [10, 3, 7, 7, Yes], [10, 4, 6, 6, Yes], [10, 5, 5, 5, Yes], [10, 6, 4, 4, Yes], [10, 7, 3, 3, Yes], [10, 8, 2, 2, Yes], [10, 9, 1, 1, Yes], [10, 10, 0, 0, Yes], [10, 11, 19, 19, Yes], [10, 12, 18, 18, Yes], [10, 13, 17, 17, Yes], [10, 14, 16, 16, Yes], [10, 15, 15, 15, Yes], [10, 16, 14, 14, Yes], [10, 17, 13, 13, Yes], [10, 18, 12, 12, Yes], [10, 19, 11, 11, Yes], [11, 0, 9, 9, Yes], [11, 1, 8, 8, Yes], [11, 2, 7, 7, Yes], [11, 3, 6, 6, Yes], [11, 4, 5, 5, Yes], [11, 5, 4, 4, Yes], [11, 6, 3, 3, Yes], [11, 7, 2, 2, Yes], [11, 8, 1, 1, Yes], [11, 9, 0, 0, Yes], [11, 10, 19, 19, Yes], [11, 11, 18, 18, Yes], [11, 12, 17, 17, Yes], [11, 13, 16, 16, Yes], [11, 14, 15, 15, Yes], [11, 15, 14, 14, Yes], [11, 16, 13, 13, Yes], [11, 17, 12, 12, Yes], [11, 18, 11, 11, Yes], [11, 19, 10, 10, Yes], [12, 0, 8, 8, Yes], [12, 1, 7, 7, Yes], [12, 2, 6, 6, Yes], [12, 3, 5, 5, Yes], [12, 4, 4, 4, Yes], [12, 5, 3, 3, Yes], [12, 6, 2, 2, Yes], [12, 7, 1, 1, Yes], [12, 8, 0, 0, Yes], [12, 9, 19, 19, Yes], [12, 10, 18, 18, Yes], [12, 11, 17, 17, Yes], [12, 12, 16, 16, Yes], [12, 13, 15, 15, Yes], [12, 14, 14, 14, Yes], [12, 15, 13, 13, Yes], [12, 16, 12, 12, Yes], [12, 17, 11, 11, Yes], [12, 18, 10, 10, Yes], [12, 19, 9, 9, Yes], [13, 0, 7, 7, Yes], [13, 1, 6, 6, Yes], [13, 2, 5, 5, Yes], [13, 3, 4, 4, Yes], [13, 4, 3, 3, Yes], [13, 5, 2, 2, Yes], [13, 6, 1, 1, Yes], [13, 7, 0, 0, Yes], [13, 8, 19, 19, Yes], [13, 9, 18, 18, Yes], [13, 10, 17, 17, Yes], [13, 11, 16, 16, Yes], [13, 12, 15, 15, Yes], [13, 13, 14, 14, Yes], [13, 14, 13, 13, Yes], [13, 15, 12, 12, Yes], [13, 16, 11, 11, Yes], [13, 17, 10, 10, Yes], [13, 18, 9, 9, Yes], [13, 19, 8, 8, Yes], [14, 0, 6, 6, Yes], [14, 1, 5, 5, Yes], [14, 2, 4, 4, Yes], [14, 3, 3, 3, Yes], [14, 4, 2, 2, Yes], [14, 5, 1, 1, Yes], [14, 6, 0, 0, Yes], [14, 7, 19, 19, Yes], [14, 8, 18, 18, Yes], [14, 9, 17, 17, Yes], [14, 10, 16, 16, Yes], [14, 11, 15, 15, Yes], [14, 12, 14, 14, Yes], [14, 13, 13, 13, Yes], [14, 14, 12, 12, Yes], [14, 15, 11, 11, Yes], [14, 16, 10, 10, Yes], [14, 17, 9, 9, Yes], [14, 18, 8, 8, Yes], [14, 19, 7, 7, Yes], [15, 0, 5, 5, Yes], [15, 1, 4, 4, Yes], [15, 2, 3, 3, Yes], [15, 3, 2, 2, Yes], [15, 4, 1, 1, Yes], [15, 5, 0, 0, Yes], [15, 6, 19, 19, Yes], [15, 7, 18, 18, Yes], [15, 8, 17, 17, Yes], [15, 9, 16, 16, Yes], [15, 10, 15, 15, Yes], [15, 11, 14, 14, Yes], [15, 12, 13, 13, Yes], [15, 13, 12, 12, Yes], [15, 14, 11, 11, Yes], [15, 15, 10, 10, Yes], [15, 16, 9, 9, Yes], [15, 17, 8, 8, Yes], [15, 18, 7, 7, Yes], [15, 19, 6, 6, Yes], [16, 0, 4, 4, Yes], [16, 1, 3, 3, Yes], [16, 2, 2, 2, Yes], [16, 3, 1, 1, Yes], [16, 4, 0, 0, Yes], [16, 5, 19, 19, Yes], [16, 6, 18, 18, Yes], [16, 7, 17, 17, Yes], [16, 8, 16, 16, Yes], [16, 9, 15, 15, Yes], [16, 10, 14, 14, Yes], [16, 11, 13, 13, Yes], [16, 12, 12, 12, Yes], [16, 13, 11, 11, Yes], [16, 14, 10, 10, Yes], [16, 15, 9, 9, Yes], [16, 16, 8, 8, Yes], [16, 17, 7, 7, Yes], [16, 18, 6, 6, Yes], [16, 19, 5, 5, Yes], [17, 0, 3, 3, Yes], [17, 1, 2, 2, Yes], [17, 2, 1, 1, Yes], [17, 3, 0, 0, Yes], [17, 4, 19, 19, Yes], [17, 5, 18, 18, Yes], [17, 6, 17, 17, Yes], [17, 7, 16, 16, Yes], [17, 8, 15, 15, Yes], [17, 9, 14, 14, Yes], [17, 10, 13, 13, Yes], [17, 11, 12, 12, Yes], [17, 12, 11, 11, Yes], [17, 13, 10, 10, Yes], [17, 14, 9, 9, Yes], [17, 15, 8, 8, Yes], [17, 16, 7, 7, Yes], [17, 17, 6, 6, Yes], [17, 18, 5, 5, Yes], [17, 19, 4, 4, Yes], [18, 0, 2, 2, Yes], [18, 1, 1, 1, Yes], [18, 2, 0, 0, Yes], [18, 3, 19, 19, Yes], [18, 4, 18, 18, Yes], [18, 5, 17, 17, Yes], [18, 6, 16, 16, Yes], [18, 7, 15, 15, Yes], [18, 8, 14, 14, Yes], [18, 9, 13, 13, Yes], [18, 10, 12, 12, Yes], [18, 11, 11, 11, Yes], [18, 12, 10, 10, Yes], [18, 13, 9, 9, Yes], [18, 14, 8, 8, Yes], [18, 15, 7, 7, Yes], [18, 16, 6, 6, Yes], [18, 17, 5, 5, Yes], [18, 18, 4, 4, Yes], [18, 19, 3, 3, Yes], [19, 0, 1, 1, Yes], [19, 1, 0, 0, Yes], [19, 2, 19, 19, Yes], [19, 3, 18, 18, Yes], [19, 4, 17, 17, Yes], [19, 5, 16, 16, Yes], [19, 6, 15, 15, Yes], [19, 7, 14, 14, Yes], [19, 8, 13, 13, Yes], [19, 9, 12, 12, Yes], [19, 10, 11, 11, Yes], [19, 11, 10, 10, Yes], [19, 12, 9, 9, Yes], [19, 13, 8, 8, Yes], [19, 14, 7, 7, Yes], [19, 15, 6, 6, Yes], [19, 16, 5, 5, Yes], [19, 17, 4, 4, Yes], [19, 18, 3, 3, Yes], [19, 19, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (20, 20, [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 19, 19, Yes], [0, 2, 18, 18, Yes], [0, 3, 17, 17, Yes], [0, 4, 16, 16, Yes], [0, 5, 15, 15, Yes], [0, 6, 14, 14, Yes], [0, 7, 13, 13, Yes], [0, 8, 12, 12, Yes], [0, 9, 11, 11, Yes], [0, 10, 10, 10, Yes], [0, 11, 9, 9, Yes], [0, 12, 8, 8, Yes], [0, 13, 7, 7, Yes], [0, 14, 6, 6, Yes], [0, 15, 5, 5, Yes], [0, 16, 4, 4, Yes], [0, 17, 3, 3, Yes], [0, 18, 2, 2, Yes], [0, 19, 1, 1, Yes], [1, 0, 19, 19, Yes], [1, 1, 18, 18, Yes], [1, 2, 17, 17, Yes], [1, 3, 16, 16, Yes], [1, 4, 15, 15, Yes], [1, 5, 14, 14, Yes], [1, 6, 13, 13, Yes], [1, 7, 12, 12, Yes], [1, 8, 11, 11, Yes], [1, 9, 10, 10, Yes], [1, 10, 9, 9, Yes], [1, 11, 8, 8, Yes], [1, 12, 7, 7, Yes], [1, 13, 6, 6, Yes], [1, 14, 5, 5, Yes], [1, 15, 4, 4, Yes], [1, 16, 3, 3, Yes], [1, 17, 2, 2, Yes], [1, 18, 1, 1, Yes], [1, 19, 0, 0, Yes], [2, 0, 18, 18, Yes], [2, 1, 17, 17, Yes], [2, 2, 16, 16, Yes], [2, 3, 15, 15, Yes], [2, 4, 14, 14, Yes], [2, 5, 13, 13, Yes], [2, 6, 12, 12, Yes], [2, 7, 11, 11, Yes], [2, 8, 10, 10, Yes], [2, 9, 9, 9, Yes], [2, 10, 8, 8, Yes], [2, 11, 7, 7, Yes], [2, 12, 6, 6, Yes], [2, 13, 5, 5, Yes], [2, 14, 4, 4, Yes], [2, 15, 3, 3, Yes], [2, 16, 2, 2, Yes], [2, 17, 1, 1, Yes], [2, 18, 0, 0, Yes], [2, 19, 19, 19, Yes], [3, 0, 17, 17, Yes], [3, 1, 16, 16, Yes], [3, 2, 15, 15, Yes], [3, 3, 14, 14, Yes], [3, 4, 13, 13, Yes], [3, 5, 12, 12, Yes], [3, 6, 11, 11, Yes], [3, 7, 10, 10, Yes], [3, 8, 9, 9, Yes], [3, 9, 8, 8, Yes], [3, 10, 7, 7, Yes], [3, 11, 6, 6, Yes], [3, 12, 5, 5, Yes], [3, 13, 4, 4, Yes], [3, 14, 3, 3, Yes], [3, 15, 2, 2, Yes], [3, 16, 1, 1, Yes], [3, 17, 0, 0, Yes], [3, 18, 19, 19, Yes], [3, 19, 18, 18, Yes], [4, 0, 16, 16, Yes], [4, 1, 15, 15, Yes], [4, 2, 14, 14, Yes], [4, 3, 13, 13, Yes], [4, 4, 12, 12, Yes], [4, 5, 11, 11, Yes], [4, 6, 10, 10, Yes], [4, 7, 9, 9, Yes], [4, 8, 8, 8, Yes], [4, 9, 7, 7, Yes], [4, 10, 6, 6, Yes], [4, 11, 5, 5, Yes], [4, 12, 4, 4, Yes], [4, 13, 3, 3, Yes], [4, 14, 2, 2, Yes], [4, 15, 1, 1, Yes], [4, 16, 0, 0, Yes], [4, 17, 19, 19, Yes], [4, 18, 18, 18, Yes], [4, 19, 17, 17, Yes], [5, 0, 15, 15, Yes], [5, 1, 14, 14, Yes], [5, 2, 13, 13, Yes], [5, 3, 12, 12, Yes], [5, 4, 11, 11, Yes], [5, 5, 10, 10, Yes], [5, 6, 9, 9, Yes], [5, 7, 8, 8, Yes], [5, 8, 7, 7, Yes], [5, 9, 6, 6, Yes], [5, 10, 5, 5, Yes], [5, 11, 4, 4, Yes], [5, 12, 3, 3, Yes], [5, 13, 2, 2, Yes], [5, 14, 1, 1, Yes], [5, 15, 0, 0, Yes], [5, 16, 19, 19, Yes], [5, 17, 18, 18, Yes], [5, 18, 17, 17, Yes], [5, 19, 16, 16, Yes], [6, 0, 14, 14, Yes], [6, 1, 13, 13, Yes], [6, 2, 12, 12, Yes], [6, 3, 11, 11, Yes], [6, 4, 10, 10, Yes], [6, 5, 9, 9, Yes], [6, 6, 8, 8, Yes], [6, 7, 7, 7, Yes], [6, 8, 6, 6, Yes], [6, 9, 5, 5, Yes], [6, 10, 4, 4, Yes], [6, 11, 3, 3, Yes], [6, 12, 2, 2, Yes], [6, 13, 1, 1, Yes], [6, 14, 0, 0, Yes], [6, 15, 19, 19, Yes], [6, 16, 18, 18, Yes], [6, 17, 17, 17, Yes], [6, 18, 16, 16, Yes], [6, 19, 15, 15, Yes], [7, 0, 13, 13, Yes], [7, 1, 12, 12, Yes], [7, 2, 11, 11, Yes], [7, 3, 10, 10, Yes], [7, 4, 9, 9, Yes], [7, 5, 8, 8, Yes], [7, 6, 7, 7, Yes], [7, 7, 6, 6, Yes], [7, 8, 5, 5, Yes], [7, 9, 4, 4, Yes], [7, 10, 3, 3, Yes], [7, 11, 2, 2, Yes], [7, 12, 1, 1, Yes], [7, 13, 0, 0, Yes], [7, 14, 19, 19, Yes], [7, 15, 18, 18, Yes], [7, 16, 17, 17, Yes], [7, 17, 16, 16, Yes], [7, 18, 15, 15, Yes], [7, 19, 14, 14, Yes], [8, 0, 12, 12, Yes], [8, 1, 11, 11, Yes], [8, 2, 10, 10, Yes], [8, 3, 9, 9, Yes], [8, 4, 8, 8, Yes], [8, 5, 7, 7, Yes], [8, 6, 6, 6, Yes], [8, 7, 5, 5, Yes], [8, 8, 4, 4, Yes], [8, 9, 3, 3, Yes], [8, 10, 2, 2, Yes], [8, 11, 1, 1, Yes], [8, 12, 0, 0, Yes], [8, 13, 19, 19, Yes], [8, 14, 18, 18, Yes], [8, 15, 17, 17, Yes], [8, 16, 16, 16, Yes], [8, 17, 15, 15, Yes], [8, 18, 14, 14, Yes], [8, 19, 13, 13, Yes], [9, 0, 11, 11, Yes], [9, 1, 10, 10, Yes], [9, 2, 9, 9, Yes], [9, 3, 8, 8, Yes], [9, 4, 7, 7, Yes], [9, 5, 6, 6, Yes], [9, 6, 5, 5, Yes], [9, 7, 4, 4, Yes], [9, 8, 3, 3, Yes], [9, 9, 2, 2, Yes], [9, 10, 1, 1, Yes], [9, 11, 0, 0, Yes], [9, 12, 19, 19, Yes], [9, 13, 18, 18, Yes], [9, 14, 17, 17, Yes], [9, 15, 16, 16, Yes], [9, 16, 15, 15, Yes], [9, 17, 14, 14, Yes], [9, 18, 13, 13, Yes], [9, 19, 12, 12, Yes], [10, 0, 10, 10, Yes], [10, 1, 9, 9, Yes], [10, 2, 8, 8, Yes], [10, 3, 7, 7, Yes], [10, 4, 6, 6, Yes], [10, 5, 5, 5, Yes], [10, 6, 4, 4, Yes], [10, 7, 3, 3, Yes], [10, 8, 2, 2, Yes], [10, 9, 1, 1, Yes], [10, 10, 0, 0, Yes], [10, 11, 19, 19, Yes], [10, 12, 18, 18, Yes], [10, 13, 17, 17, Yes], [10, 14, 16, 16, Yes], [10, 15, 15, 15, Yes], [10, 16, 14, 14, Yes], [10, 17, 13, 13, Yes], [10, 18, 12, 12, Yes], [10, 19, 11, 11, Yes], [11, 0, 9, 9, Yes], [11, 1, 8, 8, Yes], [11, 2, 7, 7, Yes], [11, 3, 6, 6, Yes], [11, 4, 5, 5, Yes], [11, 5, 4, 4, Yes], [11, 6, 3, 3, Yes], [11, 7, 2, 2, Yes], [11, 8, 1, 1, Yes], [11, 9, 0, 0, Yes], [11, 10, 19, 19, Yes], [11, 11, 18, 18, Yes], [11, 12, 17, 17, Yes], [11, 13, 16, 16, Yes], [11, 14, 15, 15, Yes], [11, 15, 14, 14, Yes], [11, 16, 13, 13, Yes], [11, 17, 12, 12, Yes], [11, 18, 11, 11, Yes], [11, 19, 10, 10, Yes], [12, 0, 8, 8, Yes], [12, 1, 7, 7, Yes], [12, 2, 6, 6, Yes], [12, 3, 5, 5, Yes], [12, 4, 4, 4, Yes], [12, 5, 3, 3, Yes], [12, 6, 2, 2, Yes], [12, 7, 1, 1, Yes], [12, 8, 0, 0, Yes], [12, 9, 19, 19, Yes], [12, 10, 18, 18, Yes], [12, 11, 17, 17, Yes], [12, 12, 16, 16, Yes], [12, 13, 15, 15, Yes], [12, 14, 14, 14, Yes], [12, 15, 13, 13, Yes], [12, 16, 12, 12, Yes], [12, 17, 11, 11, Yes], [12, 18, 10, 10, Yes], [12, 19, 9, 9, Yes], [13, 0, 7, 7, Yes], [13, 1, 6, 6, Yes], [13, 2, 5, 5, Yes], [13, 3, 4, 4, Yes], [13, 4, 3, 3, Yes], [13, 5, 2, 2, Yes], [13, 6, 1, 1, Yes], [13, 7, 0, 0, Yes], [13, 8, 19, 19, Yes], [13, 9, 18, 18, Yes], [13, 10, 17, 17, Yes], [13, 11, 16, 16, Yes], [13, 12, 15, 15, Yes], [13, 13, 14, 14, Yes], [13, 14, 13, 13, Yes], [13, 15, 12, 12, Yes], [13, 16, 11, 11, Yes], [13, 17, 10, 10, Yes], [13, 18, 9, 9, Yes], [13, 19, 8, 8, Yes], [14, 0, 6, 6, Yes], [14, 1, 5, 5, Yes], [14, 2, 4, 4, Yes], [14, 3, 3, 3, Yes], [14, 4, 2, 2, Yes], [14, 5, 1, 1, Yes], [14, 6, 0, 0, Yes], [14, 7, 19, 19, Yes], [14, 8, 18, 18, Yes], [14, 9, 17, 17, Yes], [14, 10, 16, 16, Yes], [14, 11, 15, 15, Yes], [14, 12, 14, 14, Yes], [14, 13, 13, 13, Yes], [14, 14, 12, 12, Yes], [14, 15, 11, 11, Yes], [14, 16, 10, 10, Yes], [14, 17, 9, 9, Yes], [14, 18, 8, 8, Yes], [14, 19, 7, 7, Yes], [15, 0, 5, 5, Yes], [15, 1, 4, 4, Yes], [15, 2, 3, 3, Yes], [15, 3, 2, 2, Yes], [15, 4, 1, 1, Yes], [15, 5, 0, 0, Yes], [15, 6, 19, 19, Yes], [15, 7, 18, 18, Yes], [15, 8, 17, 17, Yes], [15, 9, 16, 16, Yes], [15, 10, 15, 15, Yes], [15, 11, 14, 14, Yes], [15, 12, 13, 13, Yes], [15, 13, 12, 12, Yes], [15, 14, 11, 11, Yes], [15, 15, 10, 10, Yes], [15, 16, 9, 9, Yes], [15, 17, 8, 8, Yes], [15, 18, 7, 7, Yes], [15, 19, 6, 6, Yes], [16, 0, 4, 4, Yes], [16, 1, 3, 3, Yes], [16, 2, 2, 2, Yes], [16, 3, 1, 1, Yes], [16, 4, 0, 0, Yes], [16, 5, 19, 19, Yes], [16, 6, 18, 18, Yes], [16, 7, 17, 17, Yes], [16, 8, 16, 16, Yes], [16, 9, 15, 15, Yes], [16, 10, 14, 14, Yes], [16, 11, 13, 13, Yes], [16, 12, 12, 12, Yes], [16, 13, 11, 11, Yes], [16, 14, 10, 10, Yes], [16, 15, 9, 9, Yes], [16, 16, 8, 8, Yes], [16, 17, 7, 7, Yes], [16, 18, 6, 6, Yes], [16, 19, 5, 5, Yes], [17, 0, 3, 3, Yes], [17, 1, 2, 2, Yes], [17, 2, 1, 1, Yes], [17, 3, 0, 0, Yes], [17, 4, 19, 19, Yes], [17, 5, 18, 18, Yes], [17, 6, 17, 17, Yes], [17, 7, 16, 16, Yes], [17, 8, 15, 15, Yes], [17, 9, 14, 14, Yes], [17, 10, 13, 13, Yes], [17, 11, 12, 12, Yes], [17, 12, 11, 11, Yes], [17, 13, 10, 10, Yes], [17, 14, 9, 9, Yes], [17, 15, 8, 8, Yes], [17, 16, 7, 7, Yes], [17, 17, 6, 6, Yes], [17, 18, 5, 5, Yes], [17, 19, 4, 4, Yes], [18, 0, 2, 2, Yes], [18, 1, 1, 1, Yes], [18, 2, 0, 0, Yes], [18, 3, 19, 19, Yes], [18, 4, 18, 18, Yes], [18, 5, 17, 17, Yes], [18, 6, 16, 16, Yes], [18, 7, 15, 15, Yes], [18, 8, 14, 14, Yes], [18, 9, 13, 13, Yes], [18, 10, 12, 12, Yes], [18, 11, 11, 11, Yes], [18, 12, 10, 10, Yes], [18, 13, 9, 9, Yes], [18, 14, 8, 8, Yes], [18, 15, 7, 7, Yes], [18, 16, 6, 6, Yes], [18, 17, 5, 5, Yes], [18, 18, 4, 4, Yes], [18, 19, 3, 3, Yes], [19, 0, 1, 1, Yes], [19, 1, 0, 0, Yes], [19, 2, 19, 19, Yes], [19, 3, 18, 18, Yes], [19, 4, 17, 17, Yes], [19, 5, 16, 16, Yes], [19, 6, 15, 15, Yes], [19, 7, 14, 14, Yes], [19, 8, 13, 13, Yes], [19, 9, 12, 12, Yes], [19, 10, 11, 11, Yes], [19, 11, 10, 10, Yes], [19, 12, 9, 9, Yes], [19, 13, 8, 8, Yes], [19, 14, 7, 7, Yes], [19, 15, 6, 6, Yes], [19, 16, 5, 5, Yes], [19, 17, 4, 4, Yes], [19, 18, 3, 3, Yes], [19, 19, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0019","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eacf9b04ab3ca261356a5935de3f0bbe2dab8d5a634ccccfbca26449da58f616","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141019,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], target identity=0, and total mapping=[0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(21, 21, [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 20, 20, Yes], [0, 2, 19, 19, Yes], [0, 3, 18, 18, Yes], [0, 4, 17, 17, Yes], [0, 5, 16, 16, Yes], [0, 6, 15, 15, Yes], [0, 7, 14, 14, Yes], [0, 8, 13, 13, Yes], [0, 9, 12, 12, Yes], [0, 10, 11, 11, Yes], [0, 11, 10, 10, Yes], [0, 12, 9, 9, Yes], [0, 13, 8, 8, Yes], [0, 14, 7, 7, Yes], [0, 15, 6, 6, Yes], [0, 16, 5, 5, Yes], [0, 17, 4, 4, Yes], [0, 18, 3, 3, Yes], [0, 19, 2, 2, Yes], [0, 20, 1, 1, Yes], [1, 0, 20, 20, Yes], [1, 1, 19, 19, Yes], [1, 2, 18, 18, Yes], [1, 3, 17, 17, Yes], [1, 4, 16, 16, Yes], [1, 5, 15, 15, Yes], [1, 6, 14, 14, Yes], [1, 7, 13, 13, Yes], [1, 8, 12, 12, Yes], [1, 9, 11, 11, Yes], [1, 10, 10, 10, Yes], [1, 11, 9, 9, Yes], [1, 12, 8, 8, Yes], [1, 13, 7, 7, Yes], [1, 14, 6, 6, Yes], [1, 15, 5, 5, Yes], [1, 16, 4, 4, Yes], [1, 17, 3, 3, Yes], [1, 18, 2, 2, Yes], [1, 19, 1, 1, Yes], [1, 20, 0, 0, Yes], [2, 0, 19, 19, Yes], [2, 1, 18, 18, Yes], [2, 2, 17, 17, Yes], [2, 3, 16, 16, Yes], [2, 4, 15, 15, Yes], [2, 5, 14, 14, Yes], [2, 6, 13, 13, Yes], [2, 7, 12, 12, Yes], [2, 8, 11, 11, Yes], [2, 9, 10, 10, Yes], [2, 10, 9, 9, Yes], [2, 11, 8, 8, Yes], [2, 12, 7, 7, Yes], [2, 13, 6, 6, Yes], [2, 14, 5, 5, Yes], [2, 15, 4, 4, Yes], [2, 16, 3, 3, Yes], [2, 17, 2, 2, Yes], [2, 18, 1, 1, Yes], [2, 19, 0, 0, Yes], [2, 20, 20, 20, Yes], [3, 0, 18, 18, Yes], [3, 1, 17, 17, Yes], [3, 2, 16, 16, Yes], [3, 3, 15, 15, Yes], [3, 4, 14, 14, Yes], [3, 5, 13, 13, Yes], [3, 6, 12, 12, Yes], [3, 7, 11, 11, Yes], [3, 8, 10, 10, Yes], [3, 9, 9, 9, Yes], [3, 10, 8, 8, Yes], [3, 11, 7, 7, Yes], [3, 12, 6, 6, Yes], [3, 13, 5, 5, Yes], [3, 14, 4, 4, Yes], [3, 15, 3, 3, Yes], [3, 16, 2, 2, Yes], [3, 17, 1, 1, Yes], [3, 18, 0, 0, Yes], [3, 19, 20, 20, Yes], [3, 20, 19, 19, Yes], [4, 0, 17, 17, Yes], [4, 1, 16, 16, Yes], [4, 2, 15, 15, Yes], [4, 3, 14, 14, Yes], [4, 4, 13, 13, Yes], [4, 5, 12, 12, Yes], [4, 6, 11, 11, Yes], [4, 7, 10, 10, Yes], [4, 8, 9, 9, Yes], [4, 9, 8, 8, Yes], [4, 10, 7, 7, Yes], [4, 11, 6, 6, Yes], [4, 12, 5, 5, Yes], [4, 13, 4, 4, Yes], [4, 14, 3, 3, Yes], [4, 15, 2, 2, Yes], [4, 16, 1, 1, Yes], [4, 17, 0, 0, Yes], [4, 18, 20, 20, Yes], [4, 19, 19, 19, Yes], [4, 20, 18, 18, Yes], [5, 0, 16, 16, Yes], [5, 1, 15, 15, Yes], [5, 2, 14, 14, Yes], [5, 3, 13, 13, Yes], [5, 4, 12, 12, Yes], [5, 5, 11, 11, Yes], [5, 6, 10, 10, Yes], [5, 7, 9, 9, Yes], [5, 8, 8, 8, Yes], [5, 9, 7, 7, Yes], [5, 10, 6, 6, Yes], [5, 11, 5, 5, Yes], [5, 12, 4, 4, Yes], [5, 13, 3, 3, Yes], [5, 14, 2, 2, Yes], [5, 15, 1, 1, Yes], [5, 16, 0, 0, Yes], [5, 17, 20, 20, Yes], [5, 18, 19, 19, Yes], [5, 19, 18, 18, Yes], [5, 20, 17, 17, Yes], [6, 0, 15, 15, Yes], [6, 1, 14, 14, Yes], [6, 2, 13, 13, Yes], [6, 3, 12, 12, Yes], [6, 4, 11, 11, Yes], [6, 5, 10, 10, Yes], [6, 6, 9, 9, Yes], [6, 7, 8, 8, Yes], [6, 8, 7, 7, Yes], [6, 9, 6, 6, Yes], [6, 10, 5, 5, Yes], [6, 11, 4, 4, Yes], [6, 12, 3, 3, Yes], [6, 13, 2, 2, Yes], [6, 14, 1, 1, Yes], [6, 15, 0, 0, Yes], [6, 16, 20, 20, Yes], [6, 17, 19, 19, Yes], [6, 18, 18, 18, Yes], [6, 19, 17, 17, Yes], [6, 20, 16, 16, Yes], [7, 0, 14, 14, Yes], [7, 1, 13, 13, Yes], [7, 2, 12, 12, Yes], [7, 3, 11, 11, Yes], [7, 4, 10, 10, Yes], [7, 5, 9, 9, Yes], [7, 6, 8, 8, Yes], [7, 7, 7, 7, Yes], [7, 8, 6, 6, Yes], [7, 9, 5, 5, Yes], [7, 10, 4, 4, Yes], [7, 11, 3, 3, Yes], [7, 12, 2, 2, Yes], [7, 13, 1, 1, Yes], [7, 14, 0, 0, Yes], [7, 15, 20, 20, Yes], [7, 16, 19, 19, Yes], [7, 17, 18, 18, Yes], [7, 18, 17, 17, Yes], [7, 19, 16, 16, Yes], [7, 20, 15, 15, Yes], [8, 0, 13, 13, Yes], [8, 1, 12, 12, Yes], [8, 2, 11, 11, Yes], [8, 3, 10, 10, Yes], [8, 4, 9, 9, Yes], [8, 5, 8, 8, Yes], [8, 6, 7, 7, Yes], [8, 7, 6, 6, Yes], [8, 8, 5, 5, Yes], [8, 9, 4, 4, Yes], [8, 10, 3, 3, Yes], [8, 11, 2, 2, Yes], [8, 12, 1, 1, Yes], [8, 13, 0, 0, Yes], [8, 14, 20, 20, Yes], [8, 15, 19, 19, Yes], [8, 16, 18, 18, Yes], [8, 17, 17, 17, Yes], [8, 18, 16, 16, Yes], [8, 19, 15, 15, Yes], [8, 20, 14, 14, Yes], [9, 0, 12, 12, Yes], [9, 1, 11, 11, Yes], [9, 2, 10, 10, Yes], [9, 3, 9, 9, Yes], [9, 4, 8, 8, Yes], [9, 5, 7, 7, Yes], [9, 6, 6, 6, Yes], [9, 7, 5, 5, Yes], [9, 8, 4, 4, Yes], [9, 9, 3, 3, Yes], [9, 10, 2, 2, Yes], [9, 11, 1, 1, Yes], [9, 12, 0, 0, Yes], [9, 13, 20, 20, Yes], [9, 14, 19, 19, Yes], [9, 15, 18, 18, Yes], [9, 16, 17, 17, Yes], [9, 17, 16, 16, Yes], [9, 18, 15, 15, Yes], [9, 19, 14, 14, Yes], [9, 20, 13, 13, Yes], [10, 0, 11, 11, Yes], [10, 1, 10, 10, Yes], [10, 2, 9, 9, Yes], [10, 3, 8, 8, Yes], [10, 4, 7, 7, Yes], [10, 5, 6, 6, Yes], [10, 6, 5, 5, Yes], [10, 7, 4, 4, Yes], [10, 8, 3, 3, Yes], [10, 9, 2, 2, Yes], [10, 10, 1, 1, Yes], [10, 11, 0, 0, Yes], [10, 12, 20, 20, Yes], [10, 13, 19, 19, Yes], [10, 14, 18, 18, Yes], [10, 15, 17, 17, Yes], [10, 16, 16, 16, Yes], [10, 17, 15, 15, Yes], [10, 18, 14, 14, Yes], [10, 19, 13, 13, Yes], [10, 20, 12, 12, Yes], [11, 0, 10, 10, Yes], [11, 1, 9, 9, Yes], [11, 2, 8, 8, Yes], [11, 3, 7, 7, Yes], [11, 4, 6, 6, Yes], [11, 5, 5, 5, Yes], [11, 6, 4, 4, Yes], [11, 7, 3, 3, Yes], [11, 8, 2, 2, Yes], [11, 9, 1, 1, Yes], [11, 10, 0, 0, Yes], [11, 11, 20, 20, Yes], [11, 12, 19, 19, Yes], [11, 13, 18, 18, Yes], [11, 14, 17, 17, Yes], [11, 15, 16, 16, Yes], [11, 16, 15, 15, Yes], [11, 17, 14, 14, Yes], [11, 18, 13, 13, Yes], [11, 19, 12, 12, Yes], [11, 20, 11, 11, Yes], [12, 0, 9, 9, Yes], [12, 1, 8, 8, Yes], [12, 2, 7, 7, Yes], [12, 3, 6, 6, Yes], [12, 4, 5, 5, Yes], [12, 5, 4, 4, Yes], [12, 6, 3, 3, Yes], [12, 7, 2, 2, Yes], [12, 8, 1, 1, Yes], [12, 9, 0, 0, Yes], [12, 10, 20, 20, Yes], [12, 11, 19, 19, Yes], [12, 12, 18, 18, Yes], [12, 13, 17, 17, Yes], [12, 14, 16, 16, Yes], [12, 15, 15, 15, Yes], [12, 16, 14, 14, Yes], [12, 17, 13, 13, Yes], [12, 18, 12, 12, Yes], [12, 19, 11, 11, Yes], [12, 20, 10, 10, Yes], [13, 0, 8, 8, Yes], [13, 1, 7, 7, Yes], [13, 2, 6, 6, Yes], [13, 3, 5, 5, Yes], [13, 4, 4, 4, Yes], [13, 5, 3, 3, Yes], [13, 6, 2, 2, Yes], [13, 7, 1, 1, Yes], [13, 8, 0, 0, Yes], [13, 9, 20, 20, Yes], [13, 10, 19, 19, Yes], [13, 11, 18, 18, Yes], [13, 12, 17, 17, Yes], [13, 13, 16, 16, Yes], [13, 14, 15, 15, Yes], [13, 15, 14, 14, Yes], [13, 16, 13, 13, Yes], [13, 17, 12, 12, Yes], [13, 18, 11, 11, Yes], [13, 19, 10, 10, Yes], [13, 20, 9, 9, Yes], [14, 0, 7, 7, Yes], [14, 1, 6, 6, Yes], [14, 2, 5, 5, Yes], [14, 3, 4, 4, Yes], [14, 4, 3, 3, Yes], [14, 5, 2, 2, Yes], [14, 6, 1, 1, Yes], [14, 7, 0, 0, Yes], [14, 8, 20, 20, Yes], [14, 9, 19, 19, Yes], [14, 10, 18, 18, Yes], [14, 11, 17, 17, Yes], [14, 12, 16, 16, Yes], [14, 13, 15, 15, Yes], [14, 14, 14, 14, Yes], [14, 15, 13, 13, Yes], [14, 16, 12, 12, Yes], [14, 17, 11, 11, Yes], [14, 18, 10, 10, Yes], [14, 19, 9, 9, Yes], [14, 20, 8, 8, Yes], [15, 0, 6, 6, Yes], [15, 1, 5, 5, Yes], [15, 2, 4, 4, Yes], [15, 3, 3, 3, Yes], [15, 4, 2, 2, Yes], [15, 5, 1, 1, Yes], [15, 6, 0, 0, Yes], [15, 7, 20, 20, Yes], [15, 8, 19, 19, Yes], [15, 9, 18, 18, Yes], [15, 10, 17, 17, Yes], [15, 11, 16, 16, Yes], [15, 12, 15, 15, Yes], [15, 13, 14, 14, Yes], [15, 14, 13, 13, Yes], [15, 15, 12, 12, Yes], [15, 16, 11, 11, Yes], [15, 17, 10, 10, Yes], [15, 18, 9, 9, Yes], [15, 19, 8, 8, Yes], [15, 20, 7, 7, Yes], [16, 0, 5, 5, Yes], [16, 1, 4, 4, Yes], [16, 2, 3, 3, Yes], [16, 3, 2, 2, Yes], [16, 4, 1, 1, Yes], [16, 5, 0, 0, Yes], [16, 6, 20, 20, Yes], [16, 7, 19, 19, Yes], [16, 8, 18, 18, Yes], [16, 9, 17, 17, Yes], [16, 10, 16, 16, Yes], [16, 11, 15, 15, Yes], [16, 12, 14, 14, Yes], [16, 13, 13, 13, Yes], [16, 14, 12, 12, Yes], [16, 15, 11, 11, Yes], [16, 16, 10, 10, Yes], [16, 17, 9, 9, Yes], [16, 18, 8, 8, Yes], [16, 19, 7, 7, Yes], [16, 20, 6, 6, Yes], [17, 0, 4, 4, Yes], [17, 1, 3, 3, Yes], [17, 2, 2, 2, Yes], [17, 3, 1, 1, Yes], [17, 4, 0, 0, Yes], [17, 5, 20, 20, Yes], [17, 6, 19, 19, Yes], [17, 7, 18, 18, Yes], [17, 8, 17, 17, Yes], [17, 9, 16, 16, Yes], [17, 10, 15, 15, Yes], [17, 11, 14, 14, Yes], [17, 12, 13, 13, Yes], [17, 13, 12, 12, Yes], [17, 14, 11, 11, Yes], [17, 15, 10, 10, Yes], [17, 16, 9, 9, Yes], [17, 17, 8, 8, Yes], [17, 18, 7, 7, Yes], [17, 19, 6, 6, Yes], [17, 20, 5, 5, Yes], [18, 0, 3, 3, Yes], [18, 1, 2, 2, Yes], [18, 2, 1, 1, Yes], [18, 3, 0, 0, Yes], [18, 4, 20, 20, Yes], [18, 5, 19, 19, Yes], [18, 6, 18, 18, Yes], [18, 7, 17, 17, Yes], [18, 8, 16, 16, Yes], [18, 9, 15, 15, Yes], [18, 10, 14, 14, Yes], [18, 11, 13, 13, Yes], [18, 12, 12, 12, Yes], [18, 13, 11, 11, Yes], [18, 14, 10, 10, Yes], [18, 15, 9, 9, Yes], [18, 16, 8, 8, Yes], [18, 17, 7, 7, Yes], [18, 18, 6, 6, Yes], [18, 19, 5, 5, Yes], [18, 20, 4, 4, Yes], [19, 0, 2, 2, Yes], [19, 1, 1, 1, Yes], [19, 2, 0, 0, Yes], [19, 3, 20, 20, Yes], [19, 4, 19, 19, Yes], [19, 5, 18, 18, Yes], [19, 6, 17, 17, Yes], [19, 7, 16, 16, Yes], [19, 8, 15, 15, Yes], [19, 9, 14, 14, Yes], [19, 10, 13, 13, Yes], [19, 11, 12, 12, Yes], [19, 12, 11, 11, Yes], [19, 13, 10, 10, Yes], [19, 14, 9, 9, Yes], [19, 15, 8, 8, Yes], [19, 16, 7, 7, Yes], [19, 17, 6, 6, Yes], [19, 18, 5, 5, Yes], [19, 19, 4, 4, Yes], [19, 20, 3, 3, Yes], [20, 0, 1, 1, Yes], [20, 1, 0, 0, Yes], [20, 2, 20, 20, Yes], [20, 3, 19, 19, Yes], [20, 4, 18, 18, Yes], [20, 5, 17, 17, Yes], [20, 6, 16, 16, Yes], [20, 7, 15, 15, Yes], [20, 8, 14, 14, Yes], [20, 9, 13, 13, Yes], [20, 10, 12, 12, Yes], [20, 11, 11, 11, Yes], [20, 12, 10, 10, Yes], [20, 13, 9, 9, Yes], [20, 14, 8, 8, Yes], [20, 15, 7, 7, Yes], [20, 16, 6, 6, Yes], [20, 17, 5, 5, Yes], [20, 18, 4, 4, Yes], [20, 19, 3, 3, Yes], [20, 20, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (21, 21, [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 20, 20, Yes], [0, 2, 19, 19, Yes], [0, 3, 18, 18, Yes], [0, 4, 17, 17, Yes], [0, 5, 16, 16, Yes], [0, 6, 15, 15, Yes], [0, 7, 14, 14, Yes], [0, 8, 13, 13, Yes], [0, 9, 12, 12, Yes], [0, 10, 11, 11, Yes], [0, 11, 10, 10, Yes], [0, 12, 9, 9, Yes], [0, 13, 8, 8, Yes], [0, 14, 7, 7, Yes], [0, 15, 6, 6, Yes], [0, 16, 5, 5, Yes], [0, 17, 4, 4, Yes], [0, 18, 3, 3, Yes], [0, 19, 2, 2, Yes], [0, 20, 1, 1, Yes], [1, 0, 20, 20, Yes], [1, 1, 19, 19, Yes], [1, 2, 18, 18, Yes], [1, 3, 17, 17, Yes], [1, 4, 16, 16, Yes], [1, 5, 15, 15, Yes], [1, 6, 14, 14, Yes], [1, 7, 13, 13, Yes], [1, 8, 12, 12, Yes], [1, 9, 11, 11, Yes], [1, 10, 10, 10, Yes], [1, 11, 9, 9, Yes], [1, 12, 8, 8, Yes], [1, 13, 7, 7, Yes], [1, 14, 6, 6, Yes], [1, 15, 5, 5, Yes], [1, 16, 4, 4, Yes], [1, 17, 3, 3, Yes], [1, 18, 2, 2, Yes], [1, 19, 1, 1, Yes], [1, 20, 0, 0, Yes], [2, 0, 19, 19, Yes], [2, 1, 18, 18, Yes], [2, 2, 17, 17, Yes], [2, 3, 16, 16, Yes], [2, 4, 15, 15, Yes], [2, 5, 14, 14, Yes], [2, 6, 13, 13, Yes], [2, 7, 12, 12, Yes], [2, 8, 11, 11, Yes], [2, 9, 10, 10, Yes], [2, 10, 9, 9, Yes], [2, 11, 8, 8, Yes], [2, 12, 7, 7, Yes], [2, 13, 6, 6, Yes], [2, 14, 5, 5, Yes], [2, 15, 4, 4, Yes], [2, 16, 3, 3, Yes], [2, 17, 2, 2, Yes], [2, 18, 1, 1, Yes], [2, 19, 0, 0, Yes], [2, 20, 20, 20, Yes], [3, 0, 18, 18, Yes], [3, 1, 17, 17, Yes], [3, 2, 16, 16, Yes], [3, 3, 15, 15, Yes], [3, 4, 14, 14, Yes], [3, 5, 13, 13, Yes], [3, 6, 12, 12, Yes], [3, 7, 11, 11, Yes], [3, 8, 10, 10, Yes], [3, 9, 9, 9, Yes], [3, 10, 8, 8, Yes], [3, 11, 7, 7, Yes], [3, 12, 6, 6, Yes], [3, 13, 5, 5, Yes], [3, 14, 4, 4, Yes], [3, 15, 3, 3, Yes], [3, 16, 2, 2, Yes], [3, 17, 1, 1, Yes], [3, 18, 0, 0, Yes], [3, 19, 20, 20, Yes], [3, 20, 19, 19, Yes], [4, 0, 17, 17, Yes], [4, 1, 16, 16, Yes], [4, 2, 15, 15, Yes], [4, 3, 14, 14, Yes], [4, 4, 13, 13, Yes], [4, 5, 12, 12, Yes], [4, 6, 11, 11, Yes], [4, 7, 10, 10, Yes], [4, 8, 9, 9, Yes], [4, 9, 8, 8, Yes], [4, 10, 7, 7, Yes], [4, 11, 6, 6, Yes], [4, 12, 5, 5, Yes], [4, 13, 4, 4, Yes], [4, 14, 3, 3, Yes], [4, 15, 2, 2, Yes], [4, 16, 1, 1, Yes], [4, 17, 0, 0, Yes], [4, 18, 20, 20, Yes], [4, 19, 19, 19, Yes], [4, 20, 18, 18, Yes], [5, 0, 16, 16, Yes], [5, 1, 15, 15, Yes], [5, 2, 14, 14, Yes], [5, 3, 13, 13, Yes], [5, 4, 12, 12, Yes], [5, 5, 11, 11, Yes], [5, 6, 10, 10, Yes], [5, 7, 9, 9, Yes], [5, 8, 8, 8, Yes], [5, 9, 7, 7, Yes], [5, 10, 6, 6, Yes], [5, 11, 5, 5, Yes], [5, 12, 4, 4, Yes], [5, 13, 3, 3, Yes], [5, 14, 2, 2, Yes], [5, 15, 1, 1, Yes], [5, 16, 0, 0, Yes], [5, 17, 20, 20, Yes], [5, 18, 19, 19, Yes], [5, 19, 18, 18, Yes], [5, 20, 17, 17, Yes], [6, 0, 15, 15, Yes], [6, 1, 14, 14, Yes], [6, 2, 13, 13, Yes], [6, 3, 12, 12, Yes], [6, 4, 11, 11, Yes], [6, 5, 10, 10, Yes], [6, 6, 9, 9, Yes], [6, 7, 8, 8, Yes], [6, 8, 7, 7, Yes], [6, 9, 6, 6, Yes], [6, 10, 5, 5, Yes], [6, 11, 4, 4, Yes], [6, 12, 3, 3, Yes], [6, 13, 2, 2, Yes], [6, 14, 1, 1, Yes], [6, 15, 0, 0, Yes], [6, 16, 20, 20, Yes], [6, 17, 19, 19, Yes], [6, 18, 18, 18, Yes], [6, 19, 17, 17, Yes], [6, 20, 16, 16, Yes], [7, 0, 14, 14, Yes], [7, 1, 13, 13, Yes], [7, 2, 12, 12, Yes], [7, 3, 11, 11, Yes], [7, 4, 10, 10, Yes], [7, 5, 9, 9, Yes], [7, 6, 8, 8, Yes], [7, 7, 7, 7, Yes], [7, 8, 6, 6, Yes], [7, 9, 5, 5, Yes], [7, 10, 4, 4, Yes], [7, 11, 3, 3, Yes], [7, 12, 2, 2, Yes], [7, 13, 1, 1, Yes], [7, 14, 0, 0, Yes], [7, 15, 20, 20, Yes], [7, 16, 19, 19, Yes], [7, 17, 18, 18, Yes], [7, 18, 17, 17, Yes], [7, 19, 16, 16, Yes], [7, 20, 15, 15, Yes], [8, 0, 13, 13, Yes], [8, 1, 12, 12, Yes], [8, 2, 11, 11, Yes], [8, 3, 10, 10, Yes], [8, 4, 9, 9, Yes], [8, 5, 8, 8, Yes], [8, 6, 7, 7, Yes], [8, 7, 6, 6, Yes], [8, 8, 5, 5, Yes], [8, 9, 4, 4, Yes], [8, 10, 3, 3, Yes], [8, 11, 2, 2, Yes], [8, 12, 1, 1, Yes], [8, 13, 0, 0, Yes], [8, 14, 20, 20, Yes], [8, 15, 19, 19, Yes], [8, 16, 18, 18, Yes], [8, 17, 17, 17, Yes], [8, 18, 16, 16, Yes], [8, 19, 15, 15, Yes], [8, 20, 14, 14, Yes], [9, 0, 12, 12, Yes], [9, 1, 11, 11, Yes], [9, 2, 10, 10, Yes], [9, 3, 9, 9, Yes], [9, 4, 8, 8, Yes], [9, 5, 7, 7, Yes], [9, 6, 6, 6, Yes], [9, 7, 5, 5, Yes], [9, 8, 4, 4, Yes], [9, 9, 3, 3, Yes], [9, 10, 2, 2, Yes], [9, 11, 1, 1, Yes], [9, 12, 0, 0, Yes], [9, 13, 20, 20, Yes], [9, 14, 19, 19, Yes], [9, 15, 18, 18, Yes], [9, 16, 17, 17, Yes], [9, 17, 16, 16, Yes], [9, 18, 15, 15, Yes], [9, 19, 14, 14, Yes], [9, 20, 13, 13, Yes], [10, 0, 11, 11, Yes], [10, 1, 10, 10, Yes], [10, 2, 9, 9, Yes], [10, 3, 8, 8, Yes], [10, 4, 7, 7, Yes], [10, 5, 6, 6, Yes], [10, 6, 5, 5, Yes], [10, 7, 4, 4, Yes], [10, 8, 3, 3, Yes], [10, 9, 2, 2, Yes], [10, 10, 1, 1, Yes], [10, 11, 0, 0, Yes], [10, 12, 20, 20, Yes], [10, 13, 19, 19, Yes], [10, 14, 18, 18, Yes], [10, 15, 17, 17, Yes], [10, 16, 16, 16, Yes], [10, 17, 15, 15, Yes], [10, 18, 14, 14, Yes], [10, 19, 13, 13, Yes], [10, 20, 12, 12, Yes], [11, 0, 10, 10, Yes], [11, 1, 9, 9, Yes], [11, 2, 8, 8, Yes], [11, 3, 7, 7, Yes], [11, 4, 6, 6, Yes], [11, 5, 5, 5, Yes], [11, 6, 4, 4, Yes], [11, 7, 3, 3, Yes], [11, 8, 2, 2, Yes], [11, 9, 1, 1, Yes], [11, 10, 0, 0, Yes], [11, 11, 20, 20, Yes], [11, 12, 19, 19, Yes], [11, 13, 18, 18, Yes], [11, 14, 17, 17, Yes], [11, 15, 16, 16, Yes], [11, 16, 15, 15, Yes], [11, 17, 14, 14, Yes], [11, 18, 13, 13, Yes], [11, 19, 12, 12, Yes], [11, 20, 11, 11, Yes], [12, 0, 9, 9, Yes], [12, 1, 8, 8, Yes], [12, 2, 7, 7, Yes], [12, 3, 6, 6, Yes], [12, 4, 5, 5, Yes], [12, 5, 4, 4, Yes], [12, 6, 3, 3, Yes], [12, 7, 2, 2, Yes], [12, 8, 1, 1, Yes], [12, 9, 0, 0, Yes], [12, 10, 20, 20, Yes], [12, 11, 19, 19, Yes], [12, 12, 18, 18, Yes], [12, 13, 17, 17, Yes], [12, 14, 16, 16, Yes], [12, 15, 15, 15, Yes], [12, 16, 14, 14, Yes], [12, 17, 13, 13, Yes], [12, 18, 12, 12, Yes], [12, 19, 11, 11, Yes], [12, 20, 10, 10, Yes], [13, 0, 8, 8, Yes], [13, 1, 7, 7, Yes], [13, 2, 6, 6, Yes], [13, 3, 5, 5, Yes], [13, 4, 4, 4, Yes], [13, 5, 3, 3, Yes], [13, 6, 2, 2, Yes], [13, 7, 1, 1, Yes], [13, 8, 0, 0, Yes], [13, 9, 20, 20, Yes], [13, 10, 19, 19, Yes], [13, 11, 18, 18, Yes], [13, 12, 17, 17, Yes], [13, 13, 16, 16, Yes], [13, 14, 15, 15, Yes], [13, 15, 14, 14, Yes], [13, 16, 13, 13, Yes], [13, 17, 12, 12, Yes], [13, 18, 11, 11, Yes], [13, 19, 10, 10, Yes], [13, 20, 9, 9, Yes], [14, 0, 7, 7, Yes], [14, 1, 6, 6, Yes], [14, 2, 5, 5, Yes], [14, 3, 4, 4, Yes], [14, 4, 3, 3, Yes], [14, 5, 2, 2, Yes], [14, 6, 1, 1, Yes], [14, 7, 0, 0, Yes], [14, 8, 20, 20, Yes], [14, 9, 19, 19, Yes], [14, 10, 18, 18, Yes], [14, 11, 17, 17, Yes], [14, 12, 16, 16, Yes], [14, 13, 15, 15, Yes], [14, 14, 14, 14, Yes], [14, 15, 13, 13, Yes], [14, 16, 12, 12, Yes], [14, 17, 11, 11, Yes], [14, 18, 10, 10, Yes], [14, 19, 9, 9, Yes], [14, 20, 8, 8, Yes], [15, 0, 6, 6, Yes], [15, 1, 5, 5, Yes], [15, 2, 4, 4, Yes], [15, 3, 3, 3, Yes], [15, 4, 2, 2, Yes], [15, 5, 1, 1, Yes], [15, 6, 0, 0, Yes], [15, 7, 20, 20, Yes], [15, 8, 19, 19, Yes], [15, 9, 18, 18, Yes], [15, 10, 17, 17, Yes], [15, 11, 16, 16, Yes], [15, 12, 15, 15, Yes], [15, 13, 14, 14, Yes], [15, 14, 13, 13, Yes], [15, 15, 12, 12, Yes], [15, 16, 11, 11, Yes], [15, 17, 10, 10, Yes], [15, 18, 9, 9, Yes], [15, 19, 8, 8, Yes], [15, 20, 7, 7, Yes], [16, 0, 5, 5, Yes], [16, 1, 4, 4, Yes], [16, 2, 3, 3, Yes], [16, 3, 2, 2, Yes], [16, 4, 1, 1, Yes], [16, 5, 0, 0, Yes], [16, 6, 20, 20, Yes], [16, 7, 19, 19, Yes], [16, 8, 18, 18, Yes], [16, 9, 17, 17, Yes], [16, 10, 16, 16, Yes], [16, 11, 15, 15, Yes], [16, 12, 14, 14, Yes], [16, 13, 13, 13, Yes], [16, 14, 12, 12, Yes], [16, 15, 11, 11, Yes], [16, 16, 10, 10, Yes], [16, 17, 9, 9, Yes], [16, 18, 8, 8, Yes], [16, 19, 7, 7, Yes], [16, 20, 6, 6, Yes], [17, 0, 4, 4, Yes], [17, 1, 3, 3, Yes], [17, 2, 2, 2, Yes], [17, 3, 1, 1, Yes], [17, 4, 0, 0, Yes], [17, 5, 20, 20, Yes], [17, 6, 19, 19, Yes], [17, 7, 18, 18, Yes], [17, 8, 17, 17, Yes], [17, 9, 16, 16, Yes], [17, 10, 15, 15, Yes], [17, 11, 14, 14, Yes], [17, 12, 13, 13, Yes], [17, 13, 12, 12, Yes], [17, 14, 11, 11, Yes], [17, 15, 10, 10, Yes], [17, 16, 9, 9, Yes], [17, 17, 8, 8, Yes], [17, 18, 7, 7, Yes], [17, 19, 6, 6, Yes], [17, 20, 5, 5, Yes], [18, 0, 3, 3, Yes], [18, 1, 2, 2, Yes], [18, 2, 1, 1, Yes], [18, 3, 0, 0, Yes], [18, 4, 20, 20, Yes], [18, 5, 19, 19, Yes], [18, 6, 18, 18, Yes], [18, 7, 17, 17, Yes], [18, 8, 16, 16, Yes], [18, 9, 15, 15, Yes], [18, 10, 14, 14, Yes], [18, 11, 13, 13, Yes], [18, 12, 12, 12, Yes], [18, 13, 11, 11, Yes], [18, 14, 10, 10, Yes], [18, 15, 9, 9, Yes], [18, 16, 8, 8, Yes], [18, 17, 7, 7, Yes], [18, 18, 6, 6, Yes], [18, 19, 5, 5, Yes], [18, 20, 4, 4, Yes], [19, 0, 2, 2, Yes], [19, 1, 1, 1, Yes], [19, 2, 0, 0, Yes], [19, 3, 20, 20, Yes], [19, 4, 19, 19, Yes], [19, 5, 18, 18, Yes], [19, 6, 17, 17, Yes], [19, 7, 16, 16, Yes], [19, 8, 15, 15, Yes], [19, 9, 14, 14, Yes], [19, 10, 13, 13, Yes], [19, 11, 12, 12, Yes], [19, 12, 11, 11, Yes], [19, 13, 10, 10, Yes], [19, 14, 9, 9, Yes], [19, 15, 8, 8, Yes], [19, 16, 7, 7, Yes], [19, 17, 6, 6, Yes], [19, 18, 5, 5, Yes], [19, 19, 4, 4, Yes], [19, 20, 3, 3, Yes], [20, 0, 1, 1, Yes], [20, 1, 0, 0, Yes], [20, 2, 20, 20, Yes], [20, 3, 19, 19, Yes], [20, 4, 18, 18, Yes], [20, 5, 17, 17, Yes], [20, 6, 16, 16, Yes], [20, 7, 15, 15, Yes], [20, 8, 14, 14, Yes], [20, 9, 13, 13, Yes], [20, 10, 12, 12, Yes], [20, 11, 11, 11, Yes], [20, 12, 10, 10, Yes], [20, 13, 9, 9, Yes], [20, 14, 8, 8, Yes], [20, 15, 7, 7, Yes], [20, 16, 6, 6, Yes], [20, 17, 5, 5, Yes], [20, 18, 4, 4, Yes], [20, 19, 3, 3, Yes], [20, 20, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0020","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d82a75149208e68e0925e1bc617b3aa8163f4b60f44c3bdbf323e0ce832a63c","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141020,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], target identity=0, and total mapping=[0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(22, 22, [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 21, 21, Yes], [0, 2, 20, 20, Yes], [0, 3, 19, 19, Yes], [0, 4, 18, 18, Yes], [0, 5, 17, 17, Yes], [0, 6, 16, 16, Yes], [0, 7, 15, 15, Yes], [0, 8, 14, 14, Yes], [0, 9, 13, 13, Yes], [0, 10, 12, 12, Yes], [0, 11, 11, 11, Yes], [0, 12, 10, 10, Yes], [0, 13, 9, 9, Yes], [0, 14, 8, 8, Yes], [0, 15, 7, 7, Yes], [0, 16, 6, 6, Yes], [0, 17, 5, 5, Yes], [0, 18, 4, 4, Yes], [0, 19, 3, 3, Yes], [0, 20, 2, 2, Yes], [0, 21, 1, 1, Yes], [1, 0, 21, 21, Yes], [1, 1, 20, 20, Yes], [1, 2, 19, 19, Yes], [1, 3, 18, 18, Yes], [1, 4, 17, 17, Yes], [1, 5, 16, 16, Yes], [1, 6, 15, 15, Yes], [1, 7, 14, 14, Yes], [1, 8, 13, 13, Yes], [1, 9, 12, 12, Yes], [1, 10, 11, 11, Yes], [1, 11, 10, 10, Yes], [1, 12, 9, 9, Yes], [1, 13, 8, 8, Yes], [1, 14, 7, 7, Yes], [1, 15, 6, 6, Yes], [1, 16, 5, 5, Yes], [1, 17, 4, 4, Yes], [1, 18, 3, 3, Yes], [1, 19, 2, 2, Yes], [1, 20, 1, 1, Yes], [1, 21, 0, 0, Yes], [2, 0, 20, 20, Yes], [2, 1, 19, 19, Yes], [2, 2, 18, 18, Yes], [2, 3, 17, 17, Yes], [2, 4, 16, 16, Yes], [2, 5, 15, 15, Yes], [2, 6, 14, 14, Yes], [2, 7, 13, 13, Yes], [2, 8, 12, 12, Yes], [2, 9, 11, 11, Yes], [2, 10, 10, 10, Yes], [2, 11, 9, 9, Yes], [2, 12, 8, 8, Yes], [2, 13, 7, 7, Yes], [2, 14, 6, 6, Yes], [2, 15, 5, 5, Yes], [2, 16, 4, 4, Yes], [2, 17, 3, 3, Yes], [2, 18, 2, 2, Yes], [2, 19, 1, 1, Yes], [2, 20, 0, 0, Yes], [2, 21, 21, 21, Yes], [3, 0, 19, 19, Yes], [3, 1, 18, 18, Yes], [3, 2, 17, 17, Yes], [3, 3, 16, 16, Yes], [3, 4, 15, 15, Yes], [3, 5, 14, 14, Yes], [3, 6, 13, 13, Yes], [3, 7, 12, 12, Yes], [3, 8, 11, 11, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [3, 11, 8, 8, Yes], [3, 12, 7, 7, Yes], [3, 13, 6, 6, Yes], [3, 14, 5, 5, Yes], [3, 15, 4, 4, Yes], [3, 16, 3, 3, Yes], [3, 17, 2, 2, Yes], [3, 18, 1, 1, Yes], [3, 19, 0, 0, Yes], [3, 20, 21, 21, Yes], [3, 21, 20, 20, Yes], [4, 0, 18, 18, Yes], [4, 1, 17, 17, Yes], [4, 2, 16, 16, Yes], [4, 3, 15, 15, Yes], [4, 4, 14, 14, Yes], [4, 5, 13, 13, Yes], [4, 6, 12, 12, Yes], [4, 7, 11, 11, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [4, 11, 7, 7, Yes], [4, 12, 6, 6, Yes], [4, 13, 5, 5, Yes], [4, 14, 4, 4, Yes], [4, 15, 3, 3, Yes], [4, 16, 2, 2, Yes], [4, 17, 1, 1, Yes], [4, 18, 0, 0, Yes], [4, 19, 21, 21, Yes], [4, 20, 20, 20, Yes], [4, 21, 19, 19, Yes], [5, 0, 17, 17, Yes], [5, 1, 16, 16, Yes], [5, 2, 15, 15, Yes], [5, 3, 14, 14, Yes], [5, 4, 13, 13, Yes], [5, 5, 12, 12, Yes], [5, 6, 11, 11, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [5, 11, 6, 6, Yes], [5, 12, 5, 5, Yes], [5, 13, 4, 4, Yes], [5, 14, 3, 3, Yes], [5, 15, 2, 2, Yes], [5, 16, 1, 1, Yes], [5, 17, 0, 0, Yes], [5, 18, 21, 21, Yes], [5, 19, 20, 20, Yes], [5, 20, 19, 19, Yes], [5, 21, 18, 18, Yes], [6, 0, 16, 16, Yes], [6, 1, 15, 15, Yes], [6, 2, 14, 14, Yes], [6, 3, 13, 13, Yes], [6, 4, 12, 12, Yes], [6, 5, 11, 11, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [6, 11, 5, 5, Yes], [6, 12, 4, 4, Yes], [6, 13, 3, 3, Yes], [6, 14, 2, 2, Yes], [6, 15, 1, 1, Yes], [6, 16, 0, 0, Yes], [6, 17, 21, 21, Yes], [6, 18, 20, 20, Yes], [6, 19, 19, 19, Yes], [6, 20, 18, 18, Yes], [6, 21, 17, 17, Yes], [7, 0, 15, 15, Yes], [7, 1, 14, 14, Yes], [7, 2, 13, 13, Yes], [7, 3, 12, 12, Yes], [7, 4, 11, 11, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [7, 11, 4, 4, Yes], [7, 12, 3, 3, Yes], [7, 13, 2, 2, Yes], [7, 14, 1, 1, Yes], [7, 15, 0, 0, Yes], [7, 16, 21, 21, Yes], [7, 17, 20, 20, Yes], [7, 18, 19, 19, Yes], [7, 19, 18, 18, Yes], [7, 20, 17, 17, Yes], [7, 21, 16, 16, Yes], [8, 0, 14, 14, Yes], [8, 1, 13, 13, Yes], [8, 2, 12, 12, Yes], [8, 3, 11, 11, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [8, 11, 3, 3, Yes], [8, 12, 2, 2, Yes], [8, 13, 1, 1, Yes], [8, 14, 0, 0, Yes], [8, 15, 21, 21, Yes], [8, 16, 20, 20, Yes], [8, 17, 19, 19, Yes], [8, 18, 18, 18, Yes], [8, 19, 17, 17, Yes], [8, 20, 16, 16, Yes], [8, 21, 15, 15, Yes], [9, 0, 13, 13, Yes], [9, 1, 12, 12, Yes], [9, 2, 11, 11, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [9, 11, 2, 2, Yes], [9, 12, 1, 1, Yes], [9, 13, 0, 0, Yes], [9, 14, 21, 21, Yes], [9, 15, 20, 20, Yes], [9, 16, 19, 19, Yes], [9, 17, 18, 18, Yes], [9, 18, 17, 17, Yes], [9, 19, 16, 16, Yes], [9, 20, 15, 15, Yes], [9, 21, 14, 14, Yes], [10, 0, 12, 12, Yes], [10, 1, 11, 11, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes], [10, 11, 1, 1, Yes], [10, 12, 0, 0, Yes], [10, 13, 21, 21, Yes], [10, 14, 20, 20, Yes], [10, 15, 19, 19, Yes], [10, 16, 18, 18, Yes], [10, 17, 17, 17, Yes], [10, 18, 16, 16, Yes], [10, 19, 15, 15, Yes], [10, 20, 14, 14, Yes], [10, 21, 13, 13, Yes], [11, 0, 11, 11, Yes], [11, 1, 10, 10, Yes], [11, 2, 9, 9, Yes], [11, 3, 8, 8, Yes], [11, 4, 7, 7, Yes], [11, 5, 6, 6, Yes], [11, 6, 5, 5, Yes], [11, 7, 4, 4, Yes], [11, 8, 3, 3, Yes], [11, 9, 2, 2, Yes], [11, 10, 1, 1, Yes], [11, 11, 0, 0, Yes], [11, 12, 21, 21, Yes], [11, 13, 20, 20, Yes], [11, 14, 19, 19, Yes], [11, 15, 18, 18, Yes], [11, 16, 17, 17, Yes], [11, 17, 16, 16, Yes], [11, 18, 15, 15, Yes], [11, 19, 14, 14, Yes], [11, 20, 13, 13, Yes], [11, 21, 12, 12, Yes], [12, 0, 10, 10, Yes], [12, 1, 9, 9, Yes], [12, 2, 8, 8, Yes], [12, 3, 7, 7, Yes], [12, 4, 6, 6, Yes], [12, 5, 5, 5, Yes], [12, 6, 4, 4, Yes], [12, 7, 3, 3, Yes], [12, 8, 2, 2, Yes], [12, 9, 1, 1, Yes], [12, 10, 0, 0, Yes], [12, 11, 21, 21, Yes], [12, 12, 20, 20, Yes], [12, 13, 19, 19, Yes], [12, 14, 18, 18, Yes], [12, 15, 17, 17, Yes], [12, 16, 16, 16, Yes], [12, 17, 15, 15, Yes], [12, 18, 14, 14, Yes], [12, 19, 13, 13, Yes], [12, 20, 12, 12, Yes], [12, 21, 11, 11, Yes], [13, 0, 9, 9, Yes], [13, 1, 8, 8, Yes], [13, 2, 7, 7, Yes], [13, 3, 6, 6, Yes], [13, 4, 5, 5, Yes], [13, 5, 4, 4, Yes], [13, 6, 3, 3, Yes], [13, 7, 2, 2, Yes], [13, 8, 1, 1, Yes], [13, 9, 0, 0, Yes], [13, 10, 21, 21, Yes], [13, 11, 20, 20, Yes], [13, 12, 19, 19, Yes], [13, 13, 18, 18, Yes], [13, 14, 17, 17, Yes], [13, 15, 16, 16, Yes], [13, 16, 15, 15, Yes], [13, 17, 14, 14, Yes], [13, 18, 13, 13, Yes], [13, 19, 12, 12, Yes], [13, 20, 11, 11, Yes], [13, 21, 10, 10, Yes], [14, 0, 8, 8, Yes], [14, 1, 7, 7, Yes], [14, 2, 6, 6, Yes], [14, 3, 5, 5, Yes], [14, 4, 4, 4, Yes], [14, 5, 3, 3, Yes], [14, 6, 2, 2, Yes], [14, 7, 1, 1, Yes], [14, 8, 0, 0, Yes], [14, 9, 21, 21, Yes], [14, 10, 20, 20, Yes], [14, 11, 19, 19, Yes], [14, 12, 18, 18, Yes], [14, 13, 17, 17, Yes], [14, 14, 16, 16, Yes], [14, 15, 15, 15, Yes], [14, 16, 14, 14, Yes], [14, 17, 13, 13, Yes], [14, 18, 12, 12, Yes], [14, 19, 11, 11, Yes], [14, 20, 10, 10, Yes], [14, 21, 9, 9, Yes], [15, 0, 7, 7, Yes], [15, 1, 6, 6, Yes], [15, 2, 5, 5, Yes], [15, 3, 4, 4, Yes], [15, 4, 3, 3, Yes], [15, 5, 2, 2, Yes], [15, 6, 1, 1, Yes], [15, 7, 0, 0, Yes], [15, 8, 21, 21, Yes], [15, 9, 20, 20, Yes], [15, 10, 19, 19, Yes], [15, 11, 18, 18, Yes], [15, 12, 17, 17, Yes], [15, 13, 16, 16, Yes], [15, 14, 15, 15, Yes], [15, 15, 14, 14, Yes], [15, 16, 13, 13, Yes], [15, 17, 12, 12, Yes], [15, 18, 11, 11, Yes], [15, 19, 10, 10, Yes], [15, 20, 9, 9, Yes], [15, 21, 8, 8, Yes], [16, 0, 6, 6, Yes], [16, 1, 5, 5, Yes], [16, 2, 4, 4, Yes], [16, 3, 3, 3, Yes], [16, 4, 2, 2, Yes], [16, 5, 1, 1, Yes], [16, 6, 0, 0, Yes], [16, 7, 21, 21, Yes], [16, 8, 20, 20, Yes], [16, 9, 19, 19, Yes], [16, 10, 18, 18, Yes], [16, 11, 17, 17, Yes], [16, 12, 16, 16, Yes], [16, 13, 15, 15, Yes], [16, 14, 14, 14, Yes], [16, 15, 13, 13, Yes], [16, 16, 12, 12, Yes], [16, 17, 11, 11, Yes], [16, 18, 10, 10, Yes], [16, 19, 9, 9, Yes], [16, 20, 8, 8, Yes], [16, 21, 7, 7, Yes], [17, 0, 5, 5, Yes], [17, 1, 4, 4, Yes], [17, 2, 3, 3, Yes], [17, 3, 2, 2, Yes], [17, 4, 1, 1, Yes], [17, 5, 0, 0, Yes], [17, 6, 21, 21, Yes], [17, 7, 20, 20, Yes], [17, 8, 19, 19, Yes], [17, 9, 18, 18, Yes], [17, 10, 17, 17, Yes], [17, 11, 16, 16, Yes], [17, 12, 15, 15, Yes], [17, 13, 14, 14, Yes], [17, 14, 13, 13, Yes], [17, 15, 12, 12, Yes], [17, 16, 11, 11, Yes], [17, 17, 10, 10, Yes], [17, 18, 9, 9, Yes], [17, 19, 8, 8, Yes], [17, 20, 7, 7, Yes], [17, 21, 6, 6, Yes], [18, 0, 4, 4, Yes], [18, 1, 3, 3, Yes], [18, 2, 2, 2, Yes], [18, 3, 1, 1, Yes], [18, 4, 0, 0, Yes], [18, 5, 21, 21, Yes], [18, 6, 20, 20, Yes], [18, 7, 19, 19, Yes], [18, 8, 18, 18, Yes], [18, 9, 17, 17, Yes], [18, 10, 16, 16, Yes], [18, 11, 15, 15, Yes], [18, 12, 14, 14, Yes], [18, 13, 13, 13, Yes], [18, 14, 12, 12, Yes], [18, 15, 11, 11, Yes], [18, 16, 10, 10, Yes], [18, 17, 9, 9, Yes], [18, 18, 8, 8, Yes], [18, 19, 7, 7, Yes], [18, 20, 6, 6, Yes], [18, 21, 5, 5, Yes], [19, 0, 3, 3, Yes], [19, 1, 2, 2, Yes], [19, 2, 1, 1, Yes], [19, 3, 0, 0, Yes], [19, 4, 21, 21, Yes], [19, 5, 20, 20, Yes], [19, 6, 19, 19, Yes], [19, 7, 18, 18, Yes], [19, 8, 17, 17, Yes], [19, 9, 16, 16, Yes], [19, 10, 15, 15, Yes], [19, 11, 14, 14, Yes], [19, 12, 13, 13, Yes], [19, 13, 12, 12, Yes], [19, 14, 11, 11, Yes], [19, 15, 10, 10, Yes], [19, 16, 9, 9, Yes], [19, 17, 8, 8, Yes], [19, 18, 7, 7, Yes], [19, 19, 6, 6, Yes], [19, 20, 5, 5, Yes], [19, 21, 4, 4, Yes], [20, 0, 2, 2, Yes], [20, 1, 1, 1, Yes], [20, 2, 0, 0, Yes], [20, 3, 21, 21, Yes], [20, 4, 20, 20, Yes], [20, 5, 19, 19, Yes], [20, 6, 18, 18, Yes], [20, 7, 17, 17, Yes], [20, 8, 16, 16, Yes], [20, 9, 15, 15, Yes], [20, 10, 14, 14, Yes], [20, 11, 13, 13, Yes], [20, 12, 12, 12, Yes], [20, 13, 11, 11, Yes], [20, 14, 10, 10, Yes], [20, 15, 9, 9, Yes], [20, 16, 8, 8, Yes], [20, 17, 7, 7, Yes], [20, 18, 6, 6, Yes], [20, 19, 5, 5, Yes], [20, 20, 4, 4, Yes], [20, 21, 3, 3, Yes], [21, 0, 1, 1, Yes], [21, 1, 0, 0, Yes], [21, 2, 21, 21, Yes], [21, 3, 20, 20, Yes], [21, 4, 19, 19, Yes], [21, 5, 18, 18, Yes], [21, 6, 17, 17, Yes], [21, 7, 16, 16, Yes], [21, 8, 15, 15, Yes], [21, 9, 14, 14, Yes], [21, 10, 13, 13, Yes], [21, 11, 12, 12, Yes], [21, 12, 11, 11, Yes], [21, 13, 10, 10, Yes], [21, 14, 9, 9, Yes], [21, 15, 8, 8, Yes], [21, 16, 7, 7, Yes], [21, 17, 6, 6, Yes], [21, 18, 5, 5, Yes], [21, 19, 4, 4, Yes], [21, 20, 3, 3, Yes], [21, 21, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (22, 22, [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 21, 21, Yes], [0, 2, 20, 20, Yes], [0, 3, 19, 19, Yes], [0, 4, 18, 18, Yes], [0, 5, 17, 17, Yes], [0, 6, 16, 16, Yes], [0, 7, 15, 15, Yes], [0, 8, 14, 14, Yes], [0, 9, 13, 13, Yes], [0, 10, 12, 12, Yes], [0, 11, 11, 11, Yes], [0, 12, 10, 10, Yes], [0, 13, 9, 9, Yes], [0, 14, 8, 8, Yes], [0, 15, 7, 7, Yes], [0, 16, 6, 6, Yes], [0, 17, 5, 5, Yes], [0, 18, 4, 4, Yes], [0, 19, 3, 3, Yes], [0, 20, 2, 2, Yes], [0, 21, 1, 1, Yes], [1, 0, 21, 21, Yes], [1, 1, 20, 20, Yes], [1, 2, 19, 19, Yes], [1, 3, 18, 18, Yes], [1, 4, 17, 17, Yes], [1, 5, 16, 16, Yes], [1, 6, 15, 15, Yes], [1, 7, 14, 14, Yes], [1, 8, 13, 13, Yes], [1, 9, 12, 12, Yes], [1, 10, 11, 11, Yes], [1, 11, 10, 10, Yes], [1, 12, 9, 9, Yes], [1, 13, 8, 8, Yes], [1, 14, 7, 7, Yes], [1, 15, 6, 6, Yes], [1, 16, 5, 5, Yes], [1, 17, 4, 4, Yes], [1, 18, 3, 3, Yes], [1, 19, 2, 2, Yes], [1, 20, 1, 1, Yes], [1, 21, 0, 0, Yes], [2, 0, 20, 20, Yes], [2, 1, 19, 19, Yes], [2, 2, 18, 18, Yes], [2, 3, 17, 17, Yes], [2, 4, 16, 16, Yes], [2, 5, 15, 15, Yes], [2, 6, 14, 14, Yes], [2, 7, 13, 13, Yes], [2, 8, 12, 12, Yes], [2, 9, 11, 11, Yes], [2, 10, 10, 10, Yes], [2, 11, 9, 9, Yes], [2, 12, 8, 8, Yes], [2, 13, 7, 7, Yes], [2, 14, 6, 6, Yes], [2, 15, 5, 5, Yes], [2, 16, 4, 4, Yes], [2, 17, 3, 3, Yes], [2, 18, 2, 2, Yes], [2, 19, 1, 1, Yes], [2, 20, 0, 0, Yes], [2, 21, 21, 21, Yes], [3, 0, 19, 19, Yes], [3, 1, 18, 18, Yes], [3, 2, 17, 17, Yes], [3, 3, 16, 16, Yes], [3, 4, 15, 15, Yes], [3, 5, 14, 14, Yes], [3, 6, 13, 13, Yes], [3, 7, 12, 12, Yes], [3, 8, 11, 11, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [3, 11, 8, 8, Yes], [3, 12, 7, 7, Yes], [3, 13, 6, 6, Yes], [3, 14, 5, 5, Yes], [3, 15, 4, 4, Yes], [3, 16, 3, 3, Yes], [3, 17, 2, 2, Yes], [3, 18, 1, 1, Yes], [3, 19, 0, 0, Yes], [3, 20, 21, 21, Yes], [3, 21, 20, 20, Yes], [4, 0, 18, 18, Yes], [4, 1, 17, 17, Yes], [4, 2, 16, 16, Yes], [4, 3, 15, 15, Yes], [4, 4, 14, 14, Yes], [4, 5, 13, 13, Yes], [4, 6, 12, 12, Yes], [4, 7, 11, 11, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [4, 11, 7, 7, Yes], [4, 12, 6, 6, Yes], [4, 13, 5, 5, Yes], [4, 14, 4, 4, Yes], [4, 15, 3, 3, Yes], [4, 16, 2, 2, Yes], [4, 17, 1, 1, Yes], [4, 18, 0, 0, Yes], [4, 19, 21, 21, Yes], [4, 20, 20, 20, Yes], [4, 21, 19, 19, Yes], [5, 0, 17, 17, Yes], [5, 1, 16, 16, Yes], [5, 2, 15, 15, Yes], [5, 3, 14, 14, Yes], [5, 4, 13, 13, Yes], [5, 5, 12, 12, Yes], [5, 6, 11, 11, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [5, 11, 6, 6, Yes], [5, 12, 5, 5, Yes], [5, 13, 4, 4, Yes], [5, 14, 3, 3, Yes], [5, 15, 2, 2, Yes], [5, 16, 1, 1, Yes], [5, 17, 0, 0, Yes], [5, 18, 21, 21, Yes], [5, 19, 20, 20, Yes], [5, 20, 19, 19, Yes], [5, 21, 18, 18, Yes], [6, 0, 16, 16, Yes], [6, 1, 15, 15, Yes], [6, 2, 14, 14, Yes], [6, 3, 13, 13, Yes], [6, 4, 12, 12, Yes], [6, 5, 11, 11, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [6, 11, 5, 5, Yes], [6, 12, 4, 4, Yes], [6, 13, 3, 3, Yes], [6, 14, 2, 2, Yes], [6, 15, 1, 1, Yes], [6, 16, 0, 0, Yes], [6, 17, 21, 21, Yes], [6, 18, 20, 20, Yes], [6, 19, 19, 19, Yes], [6, 20, 18, 18, Yes], [6, 21, 17, 17, Yes], [7, 0, 15, 15, Yes], [7, 1, 14, 14, Yes], [7, 2, 13, 13, Yes], [7, 3, 12, 12, Yes], [7, 4, 11, 11, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [7, 11, 4, 4, Yes], [7, 12, 3, 3, Yes], [7, 13, 2, 2, Yes], [7, 14, 1, 1, Yes], [7, 15, 0, 0, Yes], [7, 16, 21, 21, Yes], [7, 17, 20, 20, Yes], [7, 18, 19, 19, Yes], [7, 19, 18, 18, Yes], [7, 20, 17, 17, Yes], [7, 21, 16, 16, Yes], [8, 0, 14, 14, Yes], [8, 1, 13, 13, Yes], [8, 2, 12, 12, Yes], [8, 3, 11, 11, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [8, 11, 3, 3, Yes], [8, 12, 2, 2, Yes], [8, 13, 1, 1, Yes], [8, 14, 0, 0, Yes], [8, 15, 21, 21, Yes], [8, 16, 20, 20, Yes], [8, 17, 19, 19, Yes], [8, 18, 18, 18, Yes], [8, 19, 17, 17, Yes], [8, 20, 16, 16, Yes], [8, 21, 15, 15, Yes], [9, 0, 13, 13, Yes], [9, 1, 12, 12, Yes], [9, 2, 11, 11, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [9, 11, 2, 2, Yes], [9, 12, 1, 1, Yes], [9, 13, 0, 0, Yes], [9, 14, 21, 21, Yes], [9, 15, 20, 20, Yes], [9, 16, 19, 19, Yes], [9, 17, 18, 18, Yes], [9, 18, 17, 17, Yes], [9, 19, 16, 16, Yes], [9, 20, 15, 15, Yes], [9, 21, 14, 14, Yes], [10, 0, 12, 12, Yes], [10, 1, 11, 11, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes], [10, 11, 1, 1, Yes], [10, 12, 0, 0, Yes], [10, 13, 21, 21, Yes], [10, 14, 20, 20, Yes], [10, 15, 19, 19, Yes], [10, 16, 18, 18, Yes], [10, 17, 17, 17, Yes], [10, 18, 16, 16, Yes], [10, 19, 15, 15, Yes], [10, 20, 14, 14, Yes], [10, 21, 13, 13, Yes], [11, 0, 11, 11, Yes], [11, 1, 10, 10, Yes], [11, 2, 9, 9, Yes], [11, 3, 8, 8, Yes], [11, 4, 7, 7, Yes], [11, 5, 6, 6, Yes], [11, 6, 5, 5, Yes], [11, 7, 4, 4, Yes], [11, 8, 3, 3, Yes], [11, 9, 2, 2, Yes], [11, 10, 1, 1, Yes], [11, 11, 0, 0, Yes], [11, 12, 21, 21, Yes], [11, 13, 20, 20, Yes], [11, 14, 19, 19, Yes], [11, 15, 18, 18, Yes], [11, 16, 17, 17, Yes], [11, 17, 16, 16, Yes], [11, 18, 15, 15, Yes], [11, 19, 14, 14, Yes], [11, 20, 13, 13, Yes], [11, 21, 12, 12, Yes], [12, 0, 10, 10, Yes], [12, 1, 9, 9, Yes], [12, 2, 8, 8, Yes], [12, 3, 7, 7, Yes], [12, 4, 6, 6, Yes], [12, 5, 5, 5, Yes], [12, 6, 4, 4, Yes], [12, 7, 3, 3, Yes], [12, 8, 2, 2, Yes], [12, 9, 1, 1, Yes], [12, 10, 0, 0, Yes], [12, 11, 21, 21, Yes], [12, 12, 20, 20, Yes], [12, 13, 19, 19, Yes], [12, 14, 18, 18, Yes], [12, 15, 17, 17, Yes], [12, 16, 16, 16, Yes], [12, 17, 15, 15, Yes], [12, 18, 14, 14, Yes], [12, 19, 13, 13, Yes], [12, 20, 12, 12, Yes], [12, 21, 11, 11, Yes], [13, 0, 9, 9, Yes], [13, 1, 8, 8, Yes], [13, 2, 7, 7, Yes], [13, 3, 6, 6, Yes], [13, 4, 5, 5, Yes], [13, 5, 4, 4, Yes], [13, 6, 3, 3, Yes], [13, 7, 2, 2, Yes], [13, 8, 1, 1, Yes], [13, 9, 0, 0, Yes], [13, 10, 21, 21, Yes], [13, 11, 20, 20, Yes], [13, 12, 19, 19, Yes], [13, 13, 18, 18, Yes], [13, 14, 17, 17, Yes], [13, 15, 16, 16, Yes], [13, 16, 15, 15, Yes], [13, 17, 14, 14, Yes], [13, 18, 13, 13, Yes], [13, 19, 12, 12, Yes], [13, 20, 11, 11, Yes], [13, 21, 10, 10, Yes], [14, 0, 8, 8, Yes], [14, 1, 7, 7, Yes], [14, 2, 6, 6, Yes], [14, 3, 5, 5, Yes], [14, 4, 4, 4, Yes], [14, 5, 3, 3, Yes], [14, 6, 2, 2, Yes], [14, 7, 1, 1, Yes], [14, 8, 0, 0, Yes], [14, 9, 21, 21, Yes], [14, 10, 20, 20, Yes], [14, 11, 19, 19, Yes], [14, 12, 18, 18, Yes], [14, 13, 17, 17, Yes], [14, 14, 16, 16, Yes], [14, 15, 15, 15, Yes], [14, 16, 14, 14, Yes], [14, 17, 13, 13, Yes], [14, 18, 12, 12, Yes], [14, 19, 11, 11, Yes], [14, 20, 10, 10, Yes], [14, 21, 9, 9, Yes], [15, 0, 7, 7, Yes], [15, 1, 6, 6, Yes], [15, 2, 5, 5, Yes], [15, 3, 4, 4, Yes], [15, 4, 3, 3, Yes], [15, 5, 2, 2, Yes], [15, 6, 1, 1, Yes], [15, 7, 0, 0, Yes], [15, 8, 21, 21, Yes], [15, 9, 20, 20, Yes], [15, 10, 19, 19, Yes], [15, 11, 18, 18, Yes], [15, 12, 17, 17, Yes], [15, 13, 16, 16, Yes], [15, 14, 15, 15, Yes], [15, 15, 14, 14, Yes], [15, 16, 13, 13, Yes], [15, 17, 12, 12, Yes], [15, 18, 11, 11, Yes], [15, 19, 10, 10, Yes], [15, 20, 9, 9, Yes], [15, 21, 8, 8, Yes], [16, 0, 6, 6, Yes], [16, 1, 5, 5, Yes], [16, 2, 4, 4, Yes], [16, 3, 3, 3, Yes], [16, 4, 2, 2, Yes], [16, 5, 1, 1, Yes], [16, 6, 0, 0, Yes], [16, 7, 21, 21, Yes], [16, 8, 20, 20, Yes], [16, 9, 19, 19, Yes], [16, 10, 18, 18, Yes], [16, 11, 17, 17, Yes], [16, 12, 16, 16, Yes], [16, 13, 15, 15, Yes], [16, 14, 14, 14, Yes], [16, 15, 13, 13, Yes], [16, 16, 12, 12, Yes], [16, 17, 11, 11, Yes], [16, 18, 10, 10, Yes], [16, 19, 9, 9, Yes], [16, 20, 8, 8, Yes], [16, 21, 7, 7, Yes], [17, 0, 5, 5, Yes], [17, 1, 4, 4, Yes], [17, 2, 3, 3, Yes], [17, 3, 2, 2, Yes], [17, 4, 1, 1, Yes], [17, 5, 0, 0, Yes], [17, 6, 21, 21, Yes], [17, 7, 20, 20, Yes], [17, 8, 19, 19, Yes], [17, 9, 18, 18, Yes], [17, 10, 17, 17, Yes], [17, 11, 16, 16, Yes], [17, 12, 15, 15, Yes], [17, 13, 14, 14, Yes], [17, 14, 13, 13, Yes], [17, 15, 12, 12, Yes], [17, 16, 11, 11, Yes], [17, 17, 10, 10, Yes], [17, 18, 9, 9, Yes], [17, 19, 8, 8, Yes], [17, 20, 7, 7, Yes], [17, 21, 6, 6, Yes], [18, 0, 4, 4, Yes], [18, 1, 3, 3, Yes], [18, 2, 2, 2, Yes], [18, 3, 1, 1, Yes], [18, 4, 0, 0, Yes], [18, 5, 21, 21, Yes], [18, 6, 20, 20, Yes], [18, 7, 19, 19, Yes], [18, 8, 18, 18, Yes], [18, 9, 17, 17, Yes], [18, 10, 16, 16, Yes], [18, 11, 15, 15, Yes], [18, 12, 14, 14, Yes], [18, 13, 13, 13, Yes], [18, 14, 12, 12, Yes], [18, 15, 11, 11, Yes], [18, 16, 10, 10, Yes], [18, 17, 9, 9, Yes], [18, 18, 8, 8, Yes], [18, 19, 7, 7, Yes], [18, 20, 6, 6, Yes], [18, 21, 5, 5, Yes], [19, 0, 3, 3, Yes], [19, 1, 2, 2, Yes], [19, 2, 1, 1, Yes], [19, 3, 0, 0, Yes], [19, 4, 21, 21, Yes], [19, 5, 20, 20, Yes], [19, 6, 19, 19, Yes], [19, 7, 18, 18, Yes], [19, 8, 17, 17, Yes], [19, 9, 16, 16, Yes], [19, 10, 15, 15, Yes], [19, 11, 14, 14, Yes], [19, 12, 13, 13, Yes], [19, 13, 12, 12, Yes], [19, 14, 11, 11, Yes], [19, 15, 10, 10, Yes], [19, 16, 9, 9, Yes], [19, 17, 8, 8, Yes], [19, 18, 7, 7, Yes], [19, 19, 6, 6, Yes], [19, 20, 5, 5, Yes], [19, 21, 4, 4, Yes], [20, 0, 2, 2, Yes], [20, 1, 1, 1, Yes], [20, 2, 0, 0, Yes], [20, 3, 21, 21, Yes], [20, 4, 20, 20, Yes], [20, 5, 19, 19, Yes], [20, 6, 18, 18, Yes], [20, 7, 17, 17, Yes], [20, 8, 16, 16, Yes], [20, 9, 15, 15, Yes], [20, 10, 14, 14, Yes], [20, 11, 13, 13, Yes], [20, 12, 12, 12, Yes], [20, 13, 11, 11, Yes], [20, 14, 10, 10, Yes], [20, 15, 9, 9, Yes], [20, 16, 8, 8, Yes], [20, 17, 7, 7, Yes], [20, 18, 6, 6, Yes], [20, 19, 5, 5, Yes], [20, 20, 4, 4, Yes], [20, 21, 3, 3, Yes], [21, 0, 1, 1, Yes], [21, 1, 0, 0, Yes], [21, 2, 21, 21, Yes], [21, 3, 20, 20, Yes], [21, 4, 19, 19, Yes], [21, 5, 18, 18, Yes], [21, 6, 17, 17, Yes], [21, 7, 16, 16, Yes], [21, 8, 15, 15, Yes], [21, 9, 14, 14, Yes], [21, 10, 13, 13, Yes], [21, 11, 12, 12, Yes], [21, 12, 11, 11, Yes], [21, 13, 10, 10, Yes], [21, 14, 9, 9, Yes], [21, 15, 8, 8, Yes], [21, 16, 7, 7, Yes], [21, 17, 6, 6, Yes], [21, 18, 5, 5, Yes], [21, 19, 4, 4, Yes], [21, 20, 3, 3, Yes], [21, 21, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0021","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95c254e41605b356504f4c7e1d9cc0dabed367f1ddd337146828f43561dc1415","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141021,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], target identity=0, and total mapping=[0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(23, 23, [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 22, 22, Yes], [0, 2, 21, 21, Yes], [0, 3, 20, 20, Yes], [0, 4, 19, 19, Yes], [0, 5, 18, 18, Yes], [0, 6, 17, 17, Yes], [0, 7, 16, 16, Yes], [0, 8, 15, 15, Yes], [0, 9, 14, 14, Yes], [0, 10, 13, 13, Yes], [0, 11, 12, 12, Yes], [0, 12, 11, 11, Yes], [0, 13, 10, 10, Yes], [0, 14, 9, 9, Yes], [0, 15, 8, 8, Yes], [0, 16, 7, 7, Yes], [0, 17, 6, 6, Yes], [0, 18, 5, 5, Yes], [0, 19, 4, 4, Yes], [0, 20, 3, 3, Yes], [0, 21, 2, 2, Yes], [0, 22, 1, 1, Yes], [1, 0, 22, 22, Yes], [1, 1, 21, 21, Yes], [1, 2, 20, 20, Yes], [1, 3, 19, 19, Yes], [1, 4, 18, 18, Yes], [1, 5, 17, 17, Yes], [1, 6, 16, 16, Yes], [1, 7, 15, 15, Yes], [1, 8, 14, 14, Yes], [1, 9, 13, 13, Yes], [1, 10, 12, 12, Yes], [1, 11, 11, 11, Yes], [1, 12, 10, 10, Yes], [1, 13, 9, 9, Yes], [1, 14, 8, 8, Yes], [1, 15, 7, 7, Yes], [1, 16, 6, 6, Yes], [1, 17, 5, 5, Yes], [1, 18, 4, 4, Yes], [1, 19, 3, 3, Yes], [1, 20, 2, 2, Yes], [1, 21, 1, 1, Yes], [1, 22, 0, 0, Yes], [2, 0, 21, 21, Yes], [2, 1, 20, 20, Yes], [2, 2, 19, 19, Yes], [2, 3, 18, 18, Yes], [2, 4, 17, 17, Yes], [2, 5, 16, 16, Yes], [2, 6, 15, 15, Yes], [2, 7, 14, 14, Yes], [2, 8, 13, 13, Yes], [2, 9, 12, 12, Yes], [2, 10, 11, 11, Yes], [2, 11, 10, 10, Yes], [2, 12, 9, 9, Yes], [2, 13, 8, 8, Yes], [2, 14, 7, 7, Yes], [2, 15, 6, 6, Yes], [2, 16, 5, 5, Yes], [2, 17, 4, 4, Yes], [2, 18, 3, 3, Yes], [2, 19, 2, 2, Yes], [2, 20, 1, 1, Yes], [2, 21, 0, 0, Yes], [2, 22, 22, 22, Yes], [3, 0, 20, 20, Yes], [3, 1, 19, 19, Yes], [3, 2, 18, 18, Yes], [3, 3, 17, 17, Yes], [3, 4, 16, 16, Yes], [3, 5, 15, 15, Yes], [3, 6, 14, 14, Yes], [3, 7, 13, 13, Yes], [3, 8, 12, 12, Yes], [3, 9, 11, 11, Yes], [3, 10, 10, 10, Yes], [3, 11, 9, 9, Yes], [3, 12, 8, 8, Yes], [3, 13, 7, 7, Yes], [3, 14, 6, 6, Yes], [3, 15, 5, 5, Yes], [3, 16, 4, 4, Yes], [3, 17, 3, 3, Yes], [3, 18, 2, 2, Yes], [3, 19, 1, 1, Yes], [3, 20, 0, 0, Yes], [3, 21, 22, 22, Yes], [3, 22, 21, 21, Yes], [4, 0, 19, 19, Yes], [4, 1, 18, 18, Yes], [4, 2, 17, 17, Yes], [4, 3, 16, 16, Yes], [4, 4, 15, 15, Yes], [4, 5, 14, 14, Yes], [4, 6, 13, 13, Yes], [4, 7, 12, 12, Yes], [4, 8, 11, 11, Yes], [4, 9, 10, 10, Yes], [4, 10, 9, 9, Yes], [4, 11, 8, 8, Yes], [4, 12, 7, 7, Yes], [4, 13, 6, 6, Yes], [4, 14, 5, 5, Yes], [4, 15, 4, 4, Yes], [4, 16, 3, 3, Yes], [4, 17, 2, 2, Yes], [4, 18, 1, 1, Yes], [4, 19, 0, 0, Yes], [4, 20, 22, 22, Yes], [4, 21, 21, 21, Yes], [4, 22, 20, 20, Yes], [5, 0, 18, 18, Yes], [5, 1, 17, 17, Yes], [5, 2, 16, 16, Yes], [5, 3, 15, 15, Yes], [5, 4, 14, 14, Yes], [5, 5, 13, 13, Yes], [5, 6, 12, 12, Yes], [5, 7, 11, 11, Yes], [5, 8, 10, 10, Yes], [5, 9, 9, 9, Yes], [5, 10, 8, 8, Yes], [5, 11, 7, 7, Yes], [5, 12, 6, 6, Yes], [5, 13, 5, 5, Yes], [5, 14, 4, 4, Yes], [5, 15, 3, 3, Yes], [5, 16, 2, 2, Yes], [5, 17, 1, 1, Yes], [5, 18, 0, 0, Yes], [5, 19, 22, 22, Yes], [5, 20, 21, 21, Yes], [5, 21, 20, 20, Yes], [5, 22, 19, 19, Yes], [6, 0, 17, 17, Yes], [6, 1, 16, 16, Yes], [6, 2, 15, 15, Yes], [6, 3, 14, 14, Yes], [6, 4, 13, 13, Yes], [6, 5, 12, 12, Yes], [6, 6, 11, 11, Yes], [6, 7, 10, 10, Yes], [6, 8, 9, 9, Yes], [6, 9, 8, 8, Yes], [6, 10, 7, 7, Yes], [6, 11, 6, 6, Yes], [6, 12, 5, 5, Yes], [6, 13, 4, 4, Yes], [6, 14, 3, 3, Yes], [6, 15, 2, 2, Yes], [6, 16, 1, 1, Yes], [6, 17, 0, 0, Yes], [6, 18, 22, 22, Yes], [6, 19, 21, 21, Yes], [6, 20, 20, 20, Yes], [6, 21, 19, 19, Yes], [6, 22, 18, 18, Yes], [7, 0, 16, 16, Yes], [7, 1, 15, 15, Yes], [7, 2, 14, 14, Yes], [7, 3, 13, 13, Yes], [7, 4, 12, 12, Yes], [7, 5, 11, 11, Yes], [7, 6, 10, 10, Yes], [7, 7, 9, 9, Yes], [7, 8, 8, 8, Yes], [7, 9, 7, 7, Yes], [7, 10, 6, 6, Yes], [7, 11, 5, 5, Yes], [7, 12, 4, 4, Yes], [7, 13, 3, 3, Yes], [7, 14, 2, 2, Yes], [7, 15, 1, 1, Yes], [7, 16, 0, 0, Yes], [7, 17, 22, 22, Yes], [7, 18, 21, 21, Yes], [7, 19, 20, 20, Yes], [7, 20, 19, 19, Yes], [7, 21, 18, 18, Yes], [7, 22, 17, 17, Yes], [8, 0, 15, 15, Yes], [8, 1, 14, 14, Yes], [8, 2, 13, 13, Yes], [8, 3, 12, 12, Yes], [8, 4, 11, 11, Yes], [8, 5, 10, 10, Yes], [8, 6, 9, 9, Yes], [8, 7, 8, 8, Yes], [8, 8, 7, 7, Yes], [8, 9, 6, 6, Yes], [8, 10, 5, 5, Yes], [8, 11, 4, 4, Yes], [8, 12, 3, 3, Yes], [8, 13, 2, 2, Yes], [8, 14, 1, 1, Yes], [8, 15, 0, 0, Yes], [8, 16, 22, 22, Yes], [8, 17, 21, 21, Yes], [8, 18, 20, 20, Yes], [8, 19, 19, 19, Yes], [8, 20, 18, 18, Yes], [8, 21, 17, 17, Yes], [8, 22, 16, 16, Yes], [9, 0, 14, 14, Yes], [9, 1, 13, 13, Yes], [9, 2, 12, 12, Yes], [9, 3, 11, 11, Yes], [9, 4, 10, 10, Yes], [9, 5, 9, 9, Yes], [9, 6, 8, 8, Yes], [9, 7, 7, 7, Yes], [9, 8, 6, 6, Yes], [9, 9, 5, 5, Yes], [9, 10, 4, 4, Yes], [9, 11, 3, 3, Yes], [9, 12, 2, 2, Yes], [9, 13, 1, 1, Yes], [9, 14, 0, 0, Yes], [9, 15, 22, 22, Yes], [9, 16, 21, 21, Yes], [9, 17, 20, 20, Yes], [9, 18, 19, 19, Yes], [9, 19, 18, 18, Yes], [9, 20, 17, 17, Yes], [9, 21, 16, 16, Yes], [9, 22, 15, 15, Yes], [10, 0, 13, 13, Yes], [10, 1, 12, 12, Yes], [10, 2, 11, 11, Yes], [10, 3, 10, 10, Yes], [10, 4, 9, 9, Yes], [10, 5, 8, 8, Yes], [10, 6, 7, 7, Yes], [10, 7, 6, 6, Yes], [10, 8, 5, 5, Yes], [10, 9, 4, 4, Yes], [10, 10, 3, 3, Yes], [10, 11, 2, 2, Yes], [10, 12, 1, 1, Yes], [10, 13, 0, 0, Yes], [10, 14, 22, 22, Yes], [10, 15, 21, 21, Yes], [10, 16, 20, 20, Yes], [10, 17, 19, 19, Yes], [10, 18, 18, 18, Yes], [10, 19, 17, 17, Yes], [10, 20, 16, 16, Yes], [10, 21, 15, 15, Yes], [10, 22, 14, 14, Yes], [11, 0, 12, 12, Yes], [11, 1, 11, 11, Yes], [11, 2, 10, 10, Yes], [11, 3, 9, 9, Yes], [11, 4, 8, 8, Yes], [11, 5, 7, 7, Yes], [11, 6, 6, 6, Yes], [11, 7, 5, 5, Yes], [11, 8, 4, 4, Yes], [11, 9, 3, 3, Yes], [11, 10, 2, 2, Yes], [11, 11, 1, 1, Yes], [11, 12, 0, 0, Yes], [11, 13, 22, 22, Yes], [11, 14, 21, 21, Yes], [11, 15, 20, 20, Yes], [11, 16, 19, 19, Yes], [11, 17, 18, 18, Yes], [11, 18, 17, 17, Yes], [11, 19, 16, 16, Yes], [11, 20, 15, 15, Yes], [11, 21, 14, 14, Yes], [11, 22, 13, 13, Yes], [12, 0, 11, 11, Yes], [12, 1, 10, 10, Yes], [12, 2, 9, 9, Yes], [12, 3, 8, 8, Yes], [12, 4, 7, 7, Yes], [12, 5, 6, 6, Yes], [12, 6, 5, 5, Yes], [12, 7, 4, 4, Yes], [12, 8, 3, 3, Yes], [12, 9, 2, 2, Yes], [12, 10, 1, 1, Yes], [12, 11, 0, 0, Yes], [12, 12, 22, 22, Yes], [12, 13, 21, 21, Yes], [12, 14, 20, 20, Yes], [12, 15, 19, 19, Yes], [12, 16, 18, 18, Yes], [12, 17, 17, 17, Yes], [12, 18, 16, 16, Yes], [12, 19, 15, 15, Yes], [12, 20, 14, 14, Yes], [12, 21, 13, 13, Yes], [12, 22, 12, 12, Yes], [13, 0, 10, 10, Yes], [13, 1, 9, 9, Yes], [13, 2, 8, 8, Yes], [13, 3, 7, 7, Yes], [13, 4, 6, 6, Yes], [13, 5, 5, 5, Yes], [13, 6, 4, 4, Yes], [13, 7, 3, 3, Yes], [13, 8, 2, 2, Yes], [13, 9, 1, 1, Yes], [13, 10, 0, 0, Yes], [13, 11, 22, 22, Yes], [13, 12, 21, 21, Yes], [13, 13, 20, 20, Yes], [13, 14, 19, 19, Yes], [13, 15, 18, 18, Yes], [13, 16, 17, 17, Yes], [13, 17, 16, 16, Yes], [13, 18, 15, 15, Yes], [13, 19, 14, 14, Yes], [13, 20, 13, 13, Yes], [13, 21, 12, 12, Yes], [13, 22, 11, 11, Yes], [14, 0, 9, 9, Yes], [14, 1, 8, 8, Yes], [14, 2, 7, 7, Yes], [14, 3, 6, 6, Yes], [14, 4, 5, 5, Yes], [14, 5, 4, 4, Yes], [14, 6, 3, 3, Yes], [14, 7, 2, 2, Yes], [14, 8, 1, 1, Yes], [14, 9, 0, 0, Yes], [14, 10, 22, 22, Yes], [14, 11, 21, 21, Yes], [14, 12, 20, 20, Yes], [14, 13, 19, 19, Yes], [14, 14, 18, 18, Yes], [14, 15, 17, 17, Yes], [14, 16, 16, 16, Yes], [14, 17, 15, 15, Yes], [14, 18, 14, 14, Yes], [14, 19, 13, 13, Yes], [14, 20, 12, 12, Yes], [14, 21, 11, 11, Yes], [14, 22, 10, 10, Yes], [15, 0, 8, 8, Yes], [15, 1, 7, 7, Yes], [15, 2, 6, 6, Yes], [15, 3, 5, 5, Yes], [15, 4, 4, 4, Yes], [15, 5, 3, 3, Yes], [15, 6, 2, 2, Yes], [15, 7, 1, 1, Yes], [15, 8, 0, 0, Yes], [15, 9, 22, 22, Yes], [15, 10, 21, 21, Yes], [15, 11, 20, 20, Yes], [15, 12, 19, 19, Yes], [15, 13, 18, 18, Yes], [15, 14, 17, 17, Yes], [15, 15, 16, 16, Yes], [15, 16, 15, 15, Yes], [15, 17, 14, 14, Yes], [15, 18, 13, 13, Yes], [15, 19, 12, 12, Yes], [15, 20, 11, 11, Yes], [15, 21, 10, 10, Yes], [15, 22, 9, 9, Yes], [16, 0, 7, 7, Yes], [16, 1, 6, 6, Yes], [16, 2, 5, 5, Yes], [16, 3, 4, 4, Yes], [16, 4, 3, 3, Yes], [16, 5, 2, 2, Yes], [16, 6, 1, 1, Yes], [16, 7, 0, 0, Yes], [16, 8, 22, 22, Yes], [16, 9, 21, 21, Yes], [16, 10, 20, 20, Yes], [16, 11, 19, 19, Yes], [16, 12, 18, 18, Yes], [16, 13, 17, 17, Yes], [16, 14, 16, 16, Yes], [16, 15, 15, 15, Yes], [16, 16, 14, 14, Yes], [16, 17, 13, 13, Yes], [16, 18, 12, 12, Yes], [16, 19, 11, 11, Yes], [16, 20, 10, 10, Yes], [16, 21, 9, 9, Yes], [16, 22, 8, 8, Yes], [17, 0, 6, 6, Yes], [17, 1, 5, 5, Yes], [17, 2, 4, 4, Yes], [17, 3, 3, 3, Yes], [17, 4, 2, 2, Yes], [17, 5, 1, 1, Yes], [17, 6, 0, 0, Yes], [17, 7, 22, 22, Yes], [17, 8, 21, 21, Yes], [17, 9, 20, 20, Yes], [17, 10, 19, 19, Yes], [17, 11, 18, 18, Yes], [17, 12, 17, 17, Yes], [17, 13, 16, 16, Yes], [17, 14, 15, 15, Yes], [17, 15, 14, 14, Yes], [17, 16, 13, 13, Yes], [17, 17, 12, 12, Yes], [17, 18, 11, 11, Yes], [17, 19, 10, 10, Yes], [17, 20, 9, 9, Yes], [17, 21, 8, 8, Yes], [17, 22, 7, 7, Yes], [18, 0, 5, 5, Yes], [18, 1, 4, 4, Yes], [18, 2, 3, 3, Yes], [18, 3, 2, 2, Yes], [18, 4, 1, 1, Yes], [18, 5, 0, 0, Yes], [18, 6, 22, 22, Yes], [18, 7, 21, 21, Yes], [18, 8, 20, 20, Yes], [18, 9, 19, 19, Yes], [18, 10, 18, 18, Yes], [18, 11, 17, 17, Yes], [18, 12, 16, 16, Yes], [18, 13, 15, 15, Yes], [18, 14, 14, 14, Yes], [18, 15, 13, 13, Yes], [18, 16, 12, 12, Yes], [18, 17, 11, 11, Yes], [18, 18, 10, 10, Yes], [18, 19, 9, 9, Yes], [18, 20, 8, 8, Yes], [18, 21, 7, 7, Yes], [18, 22, 6, 6, Yes], [19, 0, 4, 4, Yes], [19, 1, 3, 3, Yes], [19, 2, 2, 2, Yes], [19, 3, 1, 1, Yes], [19, 4, 0, 0, Yes], [19, 5, 22, 22, Yes], [19, 6, 21, 21, Yes], [19, 7, 20, 20, Yes], [19, 8, 19, 19, Yes], [19, 9, 18, 18, Yes], [19, 10, 17, 17, Yes], [19, 11, 16, 16, Yes], [19, 12, 15, 15, Yes], [19, 13, 14, 14, Yes], [19, 14, 13, 13, Yes], [19, 15, 12, 12, Yes], [19, 16, 11, 11, Yes], [19, 17, 10, 10, Yes], [19, 18, 9, 9, Yes], [19, 19, 8, 8, Yes], [19, 20, 7, 7, Yes], [19, 21, 6, 6, Yes], [19, 22, 5, 5, Yes], [20, 0, 3, 3, Yes], [20, 1, 2, 2, Yes], [20, 2, 1, 1, Yes], [20, 3, 0, 0, Yes], [20, 4, 22, 22, Yes], [20, 5, 21, 21, Yes], [20, 6, 20, 20, Yes], [20, 7, 19, 19, Yes], [20, 8, 18, 18, Yes], [20, 9, 17, 17, Yes], [20, 10, 16, 16, Yes], [20, 11, 15, 15, Yes], [20, 12, 14, 14, Yes], [20, 13, 13, 13, Yes], [20, 14, 12, 12, Yes], [20, 15, 11, 11, Yes], [20, 16, 10, 10, Yes], [20, 17, 9, 9, Yes], [20, 18, 8, 8, Yes], [20, 19, 7, 7, Yes], [20, 20, 6, 6, Yes], [20, 21, 5, 5, Yes], [20, 22, 4, 4, Yes], [21, 0, 2, 2, Yes], [21, 1, 1, 1, Yes], [21, 2, 0, 0, Yes], [21, 3, 22, 22, Yes], [21, 4, 21, 21, Yes], [21, 5, 20, 20, Yes], [21, 6, 19, 19, Yes], [21, 7, 18, 18, Yes], [21, 8, 17, 17, Yes], [21, 9, 16, 16, Yes], [21, 10, 15, 15, Yes], [21, 11, 14, 14, Yes], [21, 12, 13, 13, Yes], [21, 13, 12, 12, Yes], [21, 14, 11, 11, Yes], [21, 15, 10, 10, Yes], [21, 16, 9, 9, Yes], [21, 17, 8, 8, Yes], [21, 18, 7, 7, Yes], [21, 19, 6, 6, Yes], [21, 20, 5, 5, Yes], [21, 21, 4, 4, Yes], [21, 22, 3, 3, Yes], [22, 0, 1, 1, Yes], [22, 1, 0, 0, Yes], [22, 2, 22, 22, Yes], [22, 3, 21, 21, Yes], [22, 4, 20, 20, Yes], [22, 5, 19, 19, Yes], [22, 6, 18, 18, Yes], [22, 7, 17, 17, Yes], [22, 8, 16, 16, Yes], [22, 9, 15, 15, Yes], [22, 10, 14, 14, Yes], [22, 11, 13, 13, Yes], [22, 12, 12, 12, Yes], [22, 13, 11, 11, Yes], [22, 14, 10, 10, Yes], [22, 15, 9, 9, Yes], [22, 16, 8, 8, Yes], [22, 17, 7, 7, Yes], [22, 18, 6, 6, Yes], [22, 19, 5, 5, Yes], [22, 20, 4, 4, Yes], [22, 21, 3, 3, Yes], [22, 22, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (23, 23, [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 22, 22, Yes], [0, 2, 21, 21, Yes], [0, 3, 20, 20, Yes], [0, 4, 19, 19, Yes], [0, 5, 18, 18, Yes], [0, 6, 17, 17, Yes], [0, 7, 16, 16, Yes], [0, 8, 15, 15, Yes], [0, 9, 14, 14, Yes], [0, 10, 13, 13, Yes], [0, 11, 12, 12, Yes], [0, 12, 11, 11, Yes], [0, 13, 10, 10, Yes], [0, 14, 9, 9, Yes], [0, 15, 8, 8, Yes], [0, 16, 7, 7, Yes], [0, 17, 6, 6, Yes], [0, 18, 5, 5, Yes], [0, 19, 4, 4, Yes], [0, 20, 3, 3, Yes], [0, 21, 2, 2, Yes], [0, 22, 1, 1, Yes], [1, 0, 22, 22, Yes], [1, 1, 21, 21, Yes], [1, 2, 20, 20, Yes], [1, 3, 19, 19, Yes], [1, 4, 18, 18, Yes], [1, 5, 17, 17, Yes], [1, 6, 16, 16, Yes], [1, 7, 15, 15, Yes], [1, 8, 14, 14, Yes], [1, 9, 13, 13, Yes], [1, 10, 12, 12, Yes], [1, 11, 11, 11, Yes], [1, 12, 10, 10, Yes], [1, 13, 9, 9, Yes], [1, 14, 8, 8, Yes], [1, 15, 7, 7, Yes], [1, 16, 6, 6, Yes], [1, 17, 5, 5, Yes], [1, 18, 4, 4, Yes], [1, 19, 3, 3, Yes], [1, 20, 2, 2, Yes], [1, 21, 1, 1, Yes], [1, 22, 0, 0, Yes], [2, 0, 21, 21, Yes], [2, 1, 20, 20, Yes], [2, 2, 19, 19, Yes], [2, 3, 18, 18, Yes], [2, 4, 17, 17, Yes], [2, 5, 16, 16, Yes], [2, 6, 15, 15, Yes], [2, 7, 14, 14, Yes], [2, 8, 13, 13, Yes], [2, 9, 12, 12, Yes], [2, 10, 11, 11, Yes], [2, 11, 10, 10, Yes], [2, 12, 9, 9, Yes], [2, 13, 8, 8, Yes], [2, 14, 7, 7, Yes], [2, 15, 6, 6, Yes], [2, 16, 5, 5, Yes], [2, 17, 4, 4, Yes], [2, 18, 3, 3, Yes], [2, 19, 2, 2, Yes], [2, 20, 1, 1, Yes], [2, 21, 0, 0, Yes], [2, 22, 22, 22, Yes], [3, 0, 20, 20, Yes], [3, 1, 19, 19, Yes], [3, 2, 18, 18, Yes], [3, 3, 17, 17, Yes], [3, 4, 16, 16, Yes], [3, 5, 15, 15, Yes], [3, 6, 14, 14, Yes], [3, 7, 13, 13, Yes], [3, 8, 12, 12, Yes], [3, 9, 11, 11, Yes], [3, 10, 10, 10, Yes], [3, 11, 9, 9, Yes], [3, 12, 8, 8, Yes], [3, 13, 7, 7, Yes], [3, 14, 6, 6, Yes], [3, 15, 5, 5, Yes], [3, 16, 4, 4, Yes], [3, 17, 3, 3, Yes], [3, 18, 2, 2, Yes], [3, 19, 1, 1, Yes], [3, 20, 0, 0, Yes], [3, 21, 22, 22, Yes], [3, 22, 21, 21, Yes], [4, 0, 19, 19, Yes], [4, 1, 18, 18, Yes], [4, 2, 17, 17, Yes], [4, 3, 16, 16, Yes], [4, 4, 15, 15, Yes], [4, 5, 14, 14, Yes], [4, 6, 13, 13, Yes], [4, 7, 12, 12, Yes], [4, 8, 11, 11, Yes], [4, 9, 10, 10, Yes], [4, 10, 9, 9, Yes], [4, 11, 8, 8, Yes], [4, 12, 7, 7, Yes], [4, 13, 6, 6, Yes], [4, 14, 5, 5, Yes], [4, 15, 4, 4, Yes], [4, 16, 3, 3, Yes], [4, 17, 2, 2, Yes], [4, 18, 1, 1, Yes], [4, 19, 0, 0, Yes], [4, 20, 22, 22, Yes], [4, 21, 21, 21, Yes], [4, 22, 20, 20, Yes], [5, 0, 18, 18, Yes], [5, 1, 17, 17, Yes], [5, 2, 16, 16, Yes], [5, 3, 15, 15, Yes], [5, 4, 14, 14, Yes], [5, 5, 13, 13, Yes], [5, 6, 12, 12, Yes], [5, 7, 11, 11, Yes], [5, 8, 10, 10, Yes], [5, 9, 9, 9, Yes], [5, 10, 8, 8, Yes], [5, 11, 7, 7, Yes], [5, 12, 6, 6, Yes], [5, 13, 5, 5, Yes], [5, 14, 4, 4, Yes], [5, 15, 3, 3, Yes], [5, 16, 2, 2, Yes], [5, 17, 1, 1, Yes], [5, 18, 0, 0, Yes], [5, 19, 22, 22, Yes], [5, 20, 21, 21, Yes], [5, 21, 20, 20, Yes], [5, 22, 19, 19, Yes], [6, 0, 17, 17, Yes], [6, 1, 16, 16, Yes], [6, 2, 15, 15, Yes], [6, 3, 14, 14, Yes], [6, 4, 13, 13, Yes], [6, 5, 12, 12, Yes], [6, 6, 11, 11, Yes], [6, 7, 10, 10, Yes], [6, 8, 9, 9, Yes], [6, 9, 8, 8, Yes], [6, 10, 7, 7, Yes], [6, 11, 6, 6, Yes], [6, 12, 5, 5, Yes], [6, 13, 4, 4, Yes], [6, 14, 3, 3, Yes], [6, 15, 2, 2, Yes], [6, 16, 1, 1, Yes], [6, 17, 0, 0, Yes], [6, 18, 22, 22, Yes], [6, 19, 21, 21, Yes], [6, 20, 20, 20, Yes], [6, 21, 19, 19, Yes], [6, 22, 18, 18, Yes], [7, 0, 16, 16, Yes], [7, 1, 15, 15, Yes], [7, 2, 14, 14, Yes], [7, 3, 13, 13, Yes], [7, 4, 12, 12, Yes], [7, 5, 11, 11, Yes], [7, 6, 10, 10, Yes], [7, 7, 9, 9, Yes], [7, 8, 8, 8, Yes], [7, 9, 7, 7, Yes], [7, 10, 6, 6, Yes], [7, 11, 5, 5, Yes], [7, 12, 4, 4, Yes], [7, 13, 3, 3, Yes], [7, 14, 2, 2, Yes], [7, 15, 1, 1, Yes], [7, 16, 0, 0, Yes], [7, 17, 22, 22, Yes], [7, 18, 21, 21, Yes], [7, 19, 20, 20, Yes], [7, 20, 19, 19, Yes], [7, 21, 18, 18, Yes], [7, 22, 17, 17, Yes], [8, 0, 15, 15, Yes], [8, 1, 14, 14, Yes], [8, 2, 13, 13, Yes], [8, 3, 12, 12, Yes], [8, 4, 11, 11, Yes], [8, 5, 10, 10, Yes], [8, 6, 9, 9, Yes], [8, 7, 8, 8, Yes], [8, 8, 7, 7, Yes], [8, 9, 6, 6, Yes], [8, 10, 5, 5, Yes], [8, 11, 4, 4, Yes], [8, 12, 3, 3, Yes], [8, 13, 2, 2, Yes], [8, 14, 1, 1, Yes], [8, 15, 0, 0, Yes], [8, 16, 22, 22, Yes], [8, 17, 21, 21, Yes], [8, 18, 20, 20, Yes], [8, 19, 19, 19, Yes], [8, 20, 18, 18, Yes], [8, 21, 17, 17, Yes], [8, 22, 16, 16, Yes], [9, 0, 14, 14, Yes], [9, 1, 13, 13, Yes], [9, 2, 12, 12, Yes], [9, 3, 11, 11, Yes], [9, 4, 10, 10, Yes], [9, 5, 9, 9, Yes], [9, 6, 8, 8, Yes], [9, 7, 7, 7, Yes], [9, 8, 6, 6, Yes], [9, 9, 5, 5, Yes], [9, 10, 4, 4, Yes], [9, 11, 3, 3, Yes], [9, 12, 2, 2, Yes], [9, 13, 1, 1, Yes], [9, 14, 0, 0, Yes], [9, 15, 22, 22, Yes], [9, 16, 21, 21, Yes], [9, 17, 20, 20, Yes], [9, 18, 19, 19, Yes], [9, 19, 18, 18, Yes], [9, 20, 17, 17, Yes], [9, 21, 16, 16, Yes], [9, 22, 15, 15, Yes], [10, 0, 13, 13, Yes], [10, 1, 12, 12, Yes], [10, 2, 11, 11, Yes], [10, 3, 10, 10, Yes], [10, 4, 9, 9, Yes], [10, 5, 8, 8, Yes], [10, 6, 7, 7, Yes], [10, 7, 6, 6, Yes], [10, 8, 5, 5, Yes], [10, 9, 4, 4, Yes], [10, 10, 3, 3, Yes], [10, 11, 2, 2, Yes], [10, 12, 1, 1, Yes], [10, 13, 0, 0, Yes], [10, 14, 22, 22, Yes], [10, 15, 21, 21, Yes], [10, 16, 20, 20, Yes], [10, 17, 19, 19, Yes], [10, 18, 18, 18, Yes], [10, 19, 17, 17, Yes], [10, 20, 16, 16, Yes], [10, 21, 15, 15, Yes], [10, 22, 14, 14, Yes], [11, 0, 12, 12, Yes], [11, 1, 11, 11, Yes], [11, 2, 10, 10, Yes], [11, 3, 9, 9, Yes], [11, 4, 8, 8, Yes], [11, 5, 7, 7, Yes], [11, 6, 6, 6, Yes], [11, 7, 5, 5, Yes], [11, 8, 4, 4, Yes], [11, 9, 3, 3, Yes], [11, 10, 2, 2, Yes], [11, 11, 1, 1, Yes], [11, 12, 0, 0, Yes], [11, 13, 22, 22, Yes], [11, 14, 21, 21, Yes], [11, 15, 20, 20, Yes], [11, 16, 19, 19, Yes], [11, 17, 18, 18, Yes], [11, 18, 17, 17, Yes], [11, 19, 16, 16, Yes], [11, 20, 15, 15, Yes], [11, 21, 14, 14, Yes], [11, 22, 13, 13, Yes], [12, 0, 11, 11, Yes], [12, 1, 10, 10, Yes], [12, 2, 9, 9, Yes], [12, 3, 8, 8, Yes], [12, 4, 7, 7, Yes], [12, 5, 6, 6, Yes], [12, 6, 5, 5, Yes], [12, 7, 4, 4, Yes], [12, 8, 3, 3, Yes], [12, 9, 2, 2, Yes], [12, 10, 1, 1, Yes], [12, 11, 0, 0, Yes], [12, 12, 22, 22, Yes], [12, 13, 21, 21, Yes], [12, 14, 20, 20, Yes], [12, 15, 19, 19, Yes], [12, 16, 18, 18, Yes], [12, 17, 17, 17, Yes], [12, 18, 16, 16, Yes], [12, 19, 15, 15, Yes], [12, 20, 14, 14, Yes], [12, 21, 13, 13, Yes], [12, 22, 12, 12, Yes], [13, 0, 10, 10, Yes], [13, 1, 9, 9, Yes], [13, 2, 8, 8, Yes], [13, 3, 7, 7, Yes], [13, 4, 6, 6, Yes], [13, 5, 5, 5, Yes], [13, 6, 4, 4, Yes], [13, 7, 3, 3, Yes], [13, 8, 2, 2, Yes], [13, 9, 1, 1, Yes], [13, 10, 0, 0, Yes], [13, 11, 22, 22, Yes], [13, 12, 21, 21, Yes], [13, 13, 20, 20, Yes], [13, 14, 19, 19, Yes], [13, 15, 18, 18, Yes], [13, 16, 17, 17, Yes], [13, 17, 16, 16, Yes], [13, 18, 15, 15, Yes], [13, 19, 14, 14, Yes], [13, 20, 13, 13, Yes], [13, 21, 12, 12, Yes], [13, 22, 11, 11, Yes], [14, 0, 9, 9, Yes], [14, 1, 8, 8, Yes], [14, 2, 7, 7, Yes], [14, 3, 6, 6, Yes], [14, 4, 5, 5, Yes], [14, 5, 4, 4, Yes], [14, 6, 3, 3, Yes], [14, 7, 2, 2, Yes], [14, 8, 1, 1, Yes], [14, 9, 0, 0, Yes], [14, 10, 22, 22, Yes], [14, 11, 21, 21, Yes], [14, 12, 20, 20, Yes], [14, 13, 19, 19, Yes], [14, 14, 18, 18, Yes], [14, 15, 17, 17, Yes], [14, 16, 16, 16, Yes], [14, 17, 15, 15, Yes], [14, 18, 14, 14, Yes], [14, 19, 13, 13, Yes], [14, 20, 12, 12, Yes], [14, 21, 11, 11, Yes], [14, 22, 10, 10, Yes], [15, 0, 8, 8, Yes], [15, 1, 7, 7, Yes], [15, 2, 6, 6, Yes], [15, 3, 5, 5, Yes], [15, 4, 4, 4, Yes], [15, 5, 3, 3, Yes], [15, 6, 2, 2, Yes], [15, 7, 1, 1, Yes], [15, 8, 0, 0, Yes], [15, 9, 22, 22, Yes], [15, 10, 21, 21, Yes], [15, 11, 20, 20, Yes], [15, 12, 19, 19, Yes], [15, 13, 18, 18, Yes], [15, 14, 17, 17, Yes], [15, 15, 16, 16, Yes], [15, 16, 15, 15, Yes], [15, 17, 14, 14, Yes], [15, 18, 13, 13, Yes], [15, 19, 12, 12, Yes], [15, 20, 11, 11, Yes], [15, 21, 10, 10, Yes], [15, 22, 9, 9, Yes], [16, 0, 7, 7, Yes], [16, 1, 6, 6, Yes], [16, 2, 5, 5, Yes], [16, 3, 4, 4, Yes], [16, 4, 3, 3, Yes], [16, 5, 2, 2, Yes], [16, 6, 1, 1, Yes], [16, 7, 0, 0, Yes], [16, 8, 22, 22, Yes], [16, 9, 21, 21, Yes], [16, 10, 20, 20, Yes], [16, 11, 19, 19, Yes], [16, 12, 18, 18, Yes], [16, 13, 17, 17, Yes], [16, 14, 16, 16, Yes], [16, 15, 15, 15, Yes], [16, 16, 14, 14, Yes], [16, 17, 13, 13, Yes], [16, 18, 12, 12, Yes], [16, 19, 11, 11, Yes], [16, 20, 10, 10, Yes], [16, 21, 9, 9, Yes], [16, 22, 8, 8, Yes], [17, 0, 6, 6, Yes], [17, 1, 5, 5, Yes], [17, 2, 4, 4, Yes], [17, 3, 3, 3, Yes], [17, 4, 2, 2, Yes], [17, 5, 1, 1, Yes], [17, 6, 0, 0, Yes], [17, 7, 22, 22, Yes], [17, 8, 21, 21, Yes], [17, 9, 20, 20, Yes], [17, 10, 19, 19, Yes], [17, 11, 18, 18, Yes], [17, 12, 17, 17, Yes], [17, 13, 16, 16, Yes], [17, 14, 15, 15, Yes], [17, 15, 14, 14, Yes], [17, 16, 13, 13, Yes], [17, 17, 12, 12, Yes], [17, 18, 11, 11, Yes], [17, 19, 10, 10, Yes], [17, 20, 9, 9, Yes], [17, 21, 8, 8, Yes], [17, 22, 7, 7, Yes], [18, 0, 5, 5, Yes], [18, 1, 4, 4, Yes], [18, 2, 3, 3, Yes], [18, 3, 2, 2, Yes], [18, 4, 1, 1, Yes], [18, 5, 0, 0, Yes], [18, 6, 22, 22, Yes], [18, 7, 21, 21, Yes], [18, 8, 20, 20, Yes], [18, 9, 19, 19, Yes], [18, 10, 18, 18, Yes], [18, 11, 17, 17, Yes], [18, 12, 16, 16, Yes], [18, 13, 15, 15, Yes], [18, 14, 14, 14, Yes], [18, 15, 13, 13, Yes], [18, 16, 12, 12, Yes], [18, 17, 11, 11, Yes], [18, 18, 10, 10, Yes], [18, 19, 9, 9, Yes], [18, 20, 8, 8, Yes], [18, 21, 7, 7, Yes], [18, 22, 6, 6, Yes], [19, 0, 4, 4, Yes], [19, 1, 3, 3, Yes], [19, 2, 2, 2, Yes], [19, 3, 1, 1, Yes], [19, 4, 0, 0, Yes], [19, 5, 22, 22, Yes], [19, 6, 21, 21, Yes], [19, 7, 20, 20, Yes], [19, 8, 19, 19, Yes], [19, 9, 18, 18, Yes], [19, 10, 17, 17, Yes], [19, 11, 16, 16, Yes], [19, 12, 15, 15, Yes], [19, 13, 14, 14, Yes], [19, 14, 13, 13, Yes], [19, 15, 12, 12, Yes], [19, 16, 11, 11, Yes], [19, 17, 10, 10, Yes], [19, 18, 9, 9, Yes], [19, 19, 8, 8, Yes], [19, 20, 7, 7, Yes], [19, 21, 6, 6, Yes], [19, 22, 5, 5, Yes], [20, 0, 3, 3, Yes], [20, 1, 2, 2, Yes], [20, 2, 1, 1, Yes], [20, 3, 0, 0, Yes], [20, 4, 22, 22, Yes], [20, 5, 21, 21, Yes], [20, 6, 20, 20, Yes], [20, 7, 19, 19, Yes], [20, 8, 18, 18, Yes], [20, 9, 17, 17, Yes], [20, 10, 16, 16, Yes], [20, 11, 15, 15, Yes], [20, 12, 14, 14, Yes], [20, 13, 13, 13, Yes], [20, 14, 12, 12, Yes], [20, 15, 11, 11, Yes], [20, 16, 10, 10, Yes], [20, 17, 9, 9, Yes], [20, 18, 8, 8, Yes], [20, 19, 7, 7, Yes], [20, 20, 6, 6, Yes], [20, 21, 5, 5, Yes], [20, 22, 4, 4, Yes], [21, 0, 2, 2, Yes], [21, 1, 1, 1, Yes], [21, 2, 0, 0, Yes], [21, 3, 22, 22, Yes], [21, 4, 21, 21, Yes], [21, 5, 20, 20, Yes], [21, 6, 19, 19, Yes], [21, 7, 18, 18, Yes], [21, 8, 17, 17, Yes], [21, 9, 16, 16, Yes], [21, 10, 15, 15, Yes], [21, 11, 14, 14, Yes], [21, 12, 13, 13, Yes], [21, 13, 12, 12, Yes], [21, 14, 11, 11, Yes], [21, 15, 10, 10, Yes], [21, 16, 9, 9, Yes], [21, 17, 8, 8, Yes], [21, 18, 7, 7, Yes], [21, 19, 6, 6, Yes], [21, 20, 5, 5, Yes], [21, 21, 4, 4, Yes], [21, 22, 3, 3, Yes], [22, 0, 1, 1, Yes], [22, 1, 0, 0, Yes], [22, 2, 22, 22, Yes], [22, 3, 21, 21, Yes], [22, 4, 20, 20, Yes], [22, 5, 19, 19, Yes], [22, 6, 18, 18, Yes], [22, 7, 17, 17, Yes], [22, 8, 16, 16, Yes], [22, 9, 15, 15, Yes], [22, 10, 14, 14, Yes], [22, 11, 13, 13, Yes], [22, 12, 12, 12, Yes], [22, 13, 11, 11, Yes], [22, 14, 10, 10, Yes], [22, 15, 9, 9, Yes], [22, 16, 8, 8, Yes], [22, 17, 7, 7, Yes], [22, 18, 6, 6, Yes], [22, 19, 5, 5, Yes], [22, 20, 4, 4, Yes], [22, 21, 3, 3, Yes], [22, 22, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0022","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c0059a37fc307cd14ec05e55ef4c41c1f19df86f4ca37670b42cd9564094f6f","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141022,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], target identity=0, and total mapping=[0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(24, 24, [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 23, 23, Yes], [0, 2, 22, 22, Yes], [0, 3, 21, 21, Yes], [0, 4, 20, 20, Yes], [0, 5, 19, 19, Yes], [0, 6, 18, 18, Yes], [0, 7, 17, 17, Yes], [0, 8, 16, 16, Yes], [0, 9, 15, 15, Yes], [0, 10, 14, 14, Yes], [0, 11, 13, 13, Yes], [0, 12, 12, 12, Yes], [0, 13, 11, 11, Yes], [0, 14, 10, 10, Yes], [0, 15, 9, 9, Yes], [0, 16, 8, 8, Yes], [0, 17, 7, 7, Yes], [0, 18, 6, 6, Yes], [0, 19, 5, 5, Yes], [0, 20, 4, 4, Yes], [0, 21, 3, 3, Yes], [0, 22, 2, 2, Yes], [0, 23, 1, 1, Yes], [1, 0, 23, 23, Yes], [1, 1, 22, 22, Yes], [1, 2, 21, 21, Yes], [1, 3, 20, 20, Yes], [1, 4, 19, 19, Yes], [1, 5, 18, 18, Yes], [1, 6, 17, 17, Yes], [1, 7, 16, 16, Yes], [1, 8, 15, 15, Yes], [1, 9, 14, 14, Yes], [1, 10, 13, 13, Yes], [1, 11, 12, 12, Yes], [1, 12, 11, 11, Yes], [1, 13, 10, 10, Yes], [1, 14, 9, 9, Yes], [1, 15, 8, 8, Yes], [1, 16, 7, 7, Yes], [1, 17, 6, 6, Yes], [1, 18, 5, 5, Yes], [1, 19, 4, 4, Yes], [1, 20, 3, 3, Yes], [1, 21, 2, 2, Yes], [1, 22, 1, 1, Yes], [1, 23, 0, 0, Yes], [2, 0, 22, 22, Yes], [2, 1, 21, 21, Yes], [2, 2, 20, 20, Yes], [2, 3, 19, 19, Yes], [2, 4, 18, 18, Yes], [2, 5, 17, 17, Yes], [2, 6, 16, 16, Yes], [2, 7, 15, 15, Yes], [2, 8, 14, 14, Yes], [2, 9, 13, 13, Yes], [2, 10, 12, 12, Yes], [2, 11, 11, 11, Yes], [2, 12, 10, 10, Yes], [2, 13, 9, 9, Yes], [2, 14, 8, 8, Yes], [2, 15, 7, 7, Yes], [2, 16, 6, 6, Yes], [2, 17, 5, 5, Yes], [2, 18, 4, 4, Yes], [2, 19, 3, 3, Yes], [2, 20, 2, 2, Yes], [2, 21, 1, 1, Yes], [2, 22, 0, 0, Yes], [2, 23, 23, 23, Yes], [3, 0, 21, 21, Yes], [3, 1, 20, 20, Yes], [3, 2, 19, 19, Yes], [3, 3, 18, 18, Yes], [3, 4, 17, 17, Yes], [3, 5, 16, 16, Yes], [3, 6, 15, 15, Yes], [3, 7, 14, 14, Yes], [3, 8, 13, 13, Yes], [3, 9, 12, 12, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [3, 12, 9, 9, Yes], [3, 13, 8, 8, Yes], [3, 14, 7, 7, Yes], [3, 15, 6, 6, Yes], [3, 16, 5, 5, Yes], [3, 17, 4, 4, Yes], [3, 18, 3, 3, Yes], [3, 19, 2, 2, Yes], [3, 20, 1, 1, Yes], [3, 21, 0, 0, Yes], [3, 22, 23, 23, Yes], [3, 23, 22, 22, Yes], [4, 0, 20, 20, Yes], [4, 1, 19, 19, Yes], [4, 2, 18, 18, Yes], [4, 3, 17, 17, Yes], [4, 4, 16, 16, Yes], [4, 5, 15, 15, Yes], [4, 6, 14, 14, Yes], [4, 7, 13, 13, Yes], [4, 8, 12, 12, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [4, 12, 8, 8, Yes], [4, 13, 7, 7, Yes], [4, 14, 6, 6, Yes], [4, 15, 5, 5, Yes], [4, 16, 4, 4, Yes], [4, 17, 3, 3, Yes], [4, 18, 2, 2, Yes], [4, 19, 1, 1, Yes], [4, 20, 0, 0, Yes], [4, 21, 23, 23, Yes], [4, 22, 22, 22, Yes], [4, 23, 21, 21, Yes], [5, 0, 19, 19, Yes], [5, 1, 18, 18, Yes], [5, 2, 17, 17, Yes], [5, 3, 16, 16, Yes], [5, 4, 15, 15, Yes], [5, 5, 14, 14, Yes], [5, 6, 13, 13, Yes], [5, 7, 12, 12, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [5, 12, 7, 7, Yes], [5, 13, 6, 6, Yes], [5, 14, 5, 5, Yes], [5, 15, 4, 4, Yes], [5, 16, 3, 3, Yes], [5, 17, 2, 2, Yes], [5, 18, 1, 1, Yes], [5, 19, 0, 0, Yes], [5, 20, 23, 23, Yes], [5, 21, 22, 22, Yes], [5, 22, 21, 21, Yes], [5, 23, 20, 20, Yes], [6, 0, 18, 18, Yes], [6, 1, 17, 17, Yes], [6, 2, 16, 16, Yes], [6, 3, 15, 15, Yes], [6, 4, 14, 14, Yes], [6, 5, 13, 13, Yes], [6, 6, 12, 12, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [6, 12, 6, 6, Yes], [6, 13, 5, 5, Yes], [6, 14, 4, 4, Yes], [6, 15, 3, 3, Yes], [6, 16, 2, 2, Yes], [6, 17, 1, 1, Yes], [6, 18, 0, 0, Yes], [6, 19, 23, 23, Yes], [6, 20, 22, 22, Yes], [6, 21, 21, 21, Yes], [6, 22, 20, 20, Yes], [6, 23, 19, 19, Yes], [7, 0, 17, 17, Yes], [7, 1, 16, 16, Yes], [7, 2, 15, 15, Yes], [7, 3, 14, 14, Yes], [7, 4, 13, 13, Yes], [7, 5, 12, 12, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [7, 12, 5, 5, Yes], [7, 13, 4, 4, Yes], [7, 14, 3, 3, Yes], [7, 15, 2, 2, Yes], [7, 16, 1, 1, Yes], [7, 17, 0, 0, Yes], [7, 18, 23, 23, Yes], [7, 19, 22, 22, Yes], [7, 20, 21, 21, Yes], [7, 21, 20, 20, Yes], [7, 22, 19, 19, Yes], [7, 23, 18, 18, Yes], [8, 0, 16, 16, Yes], [8, 1, 15, 15, Yes], [8, 2, 14, 14, Yes], [8, 3, 13, 13, Yes], [8, 4, 12, 12, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [8, 12, 4, 4, Yes], [8, 13, 3, 3, Yes], [8, 14, 2, 2, Yes], [8, 15, 1, 1, Yes], [8, 16, 0, 0, Yes], [8, 17, 23, 23, Yes], [8, 18, 22, 22, Yes], [8, 19, 21, 21, Yes], [8, 20, 20, 20, Yes], [8, 21, 19, 19, Yes], [8, 22, 18, 18, Yes], [8, 23, 17, 17, Yes], [9, 0, 15, 15, Yes], [9, 1, 14, 14, Yes], [9, 2, 13, 13, Yes], [9, 3, 12, 12, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [9, 12, 3, 3, Yes], [9, 13, 2, 2, Yes], [9, 14, 1, 1, Yes], [9, 15, 0, 0, Yes], [9, 16, 23, 23, Yes], [9, 17, 22, 22, Yes], [9, 18, 21, 21, Yes], [9, 19, 20, 20, Yes], [9, 20, 19, 19, Yes], [9, 21, 18, 18, Yes], [9, 22, 17, 17, Yes], [9, 23, 16, 16, Yes], [10, 0, 14, 14, Yes], [10, 1, 13, 13, Yes], [10, 2, 12, 12, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [10, 12, 2, 2, Yes], [10, 13, 1, 1, Yes], [10, 14, 0, 0, Yes], [10, 15, 23, 23, Yes], [10, 16, 22, 22, Yes], [10, 17, 21, 21, Yes], [10, 18, 20, 20, Yes], [10, 19, 19, 19, Yes], [10, 20, 18, 18, Yes], [10, 21, 17, 17, Yes], [10, 22, 16, 16, Yes], [10, 23, 15, 15, Yes], [11, 0, 13, 13, Yes], [11, 1, 12, 12, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes], [11, 12, 1, 1, Yes], [11, 13, 0, 0, Yes], [11, 14, 23, 23, Yes], [11, 15, 22, 22, Yes], [11, 16, 21, 21, Yes], [11, 17, 20, 20, Yes], [11, 18, 19, 19, Yes], [11, 19, 18, 18, Yes], [11, 20, 17, 17, Yes], [11, 21, 16, 16, Yes], [11, 22, 15, 15, Yes], [11, 23, 14, 14, Yes], [12, 0, 12, 12, Yes], [12, 1, 11, 11, Yes], [12, 2, 10, 10, Yes], [12, 3, 9, 9, Yes], [12, 4, 8, 8, Yes], [12, 5, 7, 7, Yes], [12, 6, 6, 6, Yes], [12, 7, 5, 5, Yes], [12, 8, 4, 4, Yes], [12, 9, 3, 3, Yes], [12, 10, 2, 2, Yes], [12, 11, 1, 1, Yes], [12, 12, 0, 0, Yes], [12, 13, 23, 23, Yes], [12, 14, 22, 22, Yes], [12, 15, 21, 21, Yes], [12, 16, 20, 20, Yes], [12, 17, 19, 19, Yes], [12, 18, 18, 18, Yes], [12, 19, 17, 17, Yes], [12, 20, 16, 16, Yes], [12, 21, 15, 15, Yes], [12, 22, 14, 14, Yes], [12, 23, 13, 13, Yes], [13, 0, 11, 11, Yes], [13, 1, 10, 10, Yes], [13, 2, 9, 9, Yes], [13, 3, 8, 8, Yes], [13, 4, 7, 7, Yes], [13, 5, 6, 6, Yes], [13, 6, 5, 5, Yes], [13, 7, 4, 4, Yes], [13, 8, 3, 3, Yes], [13, 9, 2, 2, Yes], [13, 10, 1, 1, Yes], [13, 11, 0, 0, Yes], [13, 12, 23, 23, Yes], [13, 13, 22, 22, Yes], [13, 14, 21, 21, Yes], [13, 15, 20, 20, Yes], [13, 16, 19, 19, Yes], [13, 17, 18, 18, Yes], [13, 18, 17, 17, Yes], [13, 19, 16, 16, Yes], [13, 20, 15, 15, Yes], [13, 21, 14, 14, Yes], [13, 22, 13, 13, Yes], [13, 23, 12, 12, Yes], [14, 0, 10, 10, Yes], [14, 1, 9, 9, Yes], [14, 2, 8, 8, Yes], [14, 3, 7, 7, Yes], [14, 4, 6, 6, Yes], [14, 5, 5, 5, Yes], [14, 6, 4, 4, Yes], [14, 7, 3, 3, Yes], [14, 8, 2, 2, Yes], [14, 9, 1, 1, Yes], [14, 10, 0, 0, Yes], [14, 11, 23, 23, Yes], [14, 12, 22, 22, Yes], [14, 13, 21, 21, Yes], [14, 14, 20, 20, Yes], [14, 15, 19, 19, Yes], [14, 16, 18, 18, Yes], [14, 17, 17, 17, Yes], [14, 18, 16, 16, Yes], [14, 19, 15, 15, Yes], [14, 20, 14, 14, Yes], [14, 21, 13, 13, Yes], [14, 22, 12, 12, Yes], [14, 23, 11, 11, Yes], [15, 0, 9, 9, Yes], [15, 1, 8, 8, Yes], [15, 2, 7, 7, Yes], [15, 3, 6, 6, Yes], [15, 4, 5, 5, Yes], [15, 5, 4, 4, Yes], [15, 6, 3, 3, Yes], [15, 7, 2, 2, Yes], [15, 8, 1, 1, Yes], [15, 9, 0, 0, Yes], [15, 10, 23, 23, Yes], [15, 11, 22, 22, Yes], [15, 12, 21, 21, Yes], [15, 13, 20, 20, Yes], [15, 14, 19, 19, Yes], [15, 15, 18, 18, Yes], [15, 16, 17, 17, Yes], [15, 17, 16, 16, Yes], [15, 18, 15, 15, Yes], [15, 19, 14, 14, Yes], [15, 20, 13, 13, Yes], [15, 21, 12, 12, Yes], [15, 22, 11, 11, Yes], [15, 23, 10, 10, Yes], [16, 0, 8, 8, Yes], [16, 1, 7, 7, Yes], [16, 2, 6, 6, Yes], [16, 3, 5, 5, Yes], [16, 4, 4, 4, Yes], [16, 5, 3, 3, Yes], [16, 6, 2, 2, Yes], [16, 7, 1, 1, Yes], [16, 8, 0, 0, Yes], [16, 9, 23, 23, Yes], [16, 10, 22, 22, Yes], [16, 11, 21, 21, Yes], [16, 12, 20, 20, Yes], [16, 13, 19, 19, Yes], [16, 14, 18, 18, Yes], [16, 15, 17, 17, Yes], [16, 16, 16, 16, Yes], [16, 17, 15, 15, Yes], [16, 18, 14, 14, Yes], [16, 19, 13, 13, Yes], [16, 20, 12, 12, Yes], [16, 21, 11, 11, Yes], [16, 22, 10, 10, Yes], [16, 23, 9, 9, Yes], [17, 0, 7, 7, Yes], [17, 1, 6, 6, Yes], [17, 2, 5, 5, Yes], [17, 3, 4, 4, Yes], [17, 4, 3, 3, Yes], [17, 5, 2, 2, Yes], [17, 6, 1, 1, Yes], [17, 7, 0, 0, Yes], [17, 8, 23, 23, Yes], [17, 9, 22, 22, Yes], [17, 10, 21, 21, Yes], [17, 11, 20, 20, Yes], [17, 12, 19, 19, Yes], [17, 13, 18, 18, Yes], [17, 14, 17, 17, Yes], [17, 15, 16, 16, Yes], [17, 16, 15, 15, Yes], [17, 17, 14, 14, Yes], [17, 18, 13, 13, Yes], [17, 19, 12, 12, Yes], [17, 20, 11, 11, Yes], [17, 21, 10, 10, Yes], [17, 22, 9, 9, Yes], [17, 23, 8, 8, Yes], [18, 0, 6, 6, Yes], [18, 1, 5, 5, Yes], [18, 2, 4, 4, Yes], [18, 3, 3, 3, Yes], [18, 4, 2, 2, Yes], [18, 5, 1, 1, Yes], [18, 6, 0, 0, Yes], [18, 7, 23, 23, Yes], [18, 8, 22, 22, Yes], [18, 9, 21, 21, Yes], [18, 10, 20, 20, Yes], [18, 11, 19, 19, Yes], [18, 12, 18, 18, Yes], [18, 13, 17, 17, Yes], [18, 14, 16, 16, Yes], [18, 15, 15, 15, Yes], [18, 16, 14, 14, Yes], [18, 17, 13, 13, Yes], [18, 18, 12, 12, Yes], [18, 19, 11, 11, Yes], [18, 20, 10, 10, Yes], [18, 21, 9, 9, Yes], [18, 22, 8, 8, Yes], [18, 23, 7, 7, Yes], [19, 0, 5, 5, Yes], [19, 1, 4, 4, Yes], [19, 2, 3, 3, Yes], [19, 3, 2, 2, Yes], [19, 4, 1, 1, Yes], [19, 5, 0, 0, Yes], [19, 6, 23, 23, Yes], [19, 7, 22, 22, Yes], [19, 8, 21, 21, Yes], [19, 9, 20, 20, Yes], [19, 10, 19, 19, Yes], [19, 11, 18, 18, Yes], [19, 12, 17, 17, Yes], [19, 13, 16, 16, Yes], [19, 14, 15, 15, Yes], [19, 15, 14, 14, Yes], [19, 16, 13, 13, Yes], [19, 17, 12, 12, Yes], [19, 18, 11, 11, Yes], [19, 19, 10, 10, Yes], [19, 20, 9, 9, Yes], [19, 21, 8, 8, Yes], [19, 22, 7, 7, Yes], [19, 23, 6, 6, Yes], [20, 0, 4, 4, Yes], [20, 1, 3, 3, Yes], [20, 2, 2, 2, Yes], [20, 3, 1, 1, Yes], [20, 4, 0, 0, Yes], [20, 5, 23, 23, Yes], [20, 6, 22, 22, Yes], [20, 7, 21, 21, Yes], [20, 8, 20, 20, Yes], [20, 9, 19, 19, Yes], [20, 10, 18, 18, Yes], [20, 11, 17, 17, Yes], [20, 12, 16, 16, Yes], [20, 13, 15, 15, Yes], [20, 14, 14, 14, Yes], [20, 15, 13, 13, Yes], [20, 16, 12, 12, Yes], [20, 17, 11, 11, Yes], [20, 18, 10, 10, Yes], [20, 19, 9, 9, Yes], [20, 20, 8, 8, Yes], [20, 21, 7, 7, Yes], [20, 22, 6, 6, Yes], [20, 23, 5, 5, Yes], [21, 0, 3, 3, Yes], [21, 1, 2, 2, Yes], [21, 2, 1, 1, Yes], [21, 3, 0, 0, Yes], [21, 4, 23, 23, Yes], [21, 5, 22, 22, Yes], [21, 6, 21, 21, Yes], [21, 7, 20, 20, Yes], [21, 8, 19, 19, Yes], [21, 9, 18, 18, Yes], [21, 10, 17, 17, Yes], [21, 11, 16, 16, Yes], [21, 12, 15, 15, Yes], [21, 13, 14, 14, Yes], [21, 14, 13, 13, Yes], [21, 15, 12, 12, Yes], [21, 16, 11, 11, Yes], [21, 17, 10, 10, Yes], [21, 18, 9, 9, Yes], [21, 19, 8, 8, Yes], [21, 20, 7, 7, Yes], [21, 21, 6, 6, Yes], [21, 22, 5, 5, Yes], [21, 23, 4, 4, Yes], [22, 0, 2, 2, Yes], [22, 1, 1, 1, Yes], [22, 2, 0, 0, Yes], [22, 3, 23, 23, Yes], [22, 4, 22, 22, Yes], [22, 5, 21, 21, Yes], [22, 6, 20, 20, Yes], [22, 7, 19, 19, Yes], [22, 8, 18, 18, Yes], [22, 9, 17, 17, Yes], [22, 10, 16, 16, Yes], [22, 11, 15, 15, Yes], [22, 12, 14, 14, Yes], [22, 13, 13, 13, Yes], [22, 14, 12, 12, Yes], [22, 15, 11, 11, Yes], [22, 16, 10, 10, Yes], [22, 17, 9, 9, Yes], [22, 18, 8, 8, Yes], [22, 19, 7, 7, Yes], [22, 20, 6, 6, Yes], [22, 21, 5, 5, Yes], [22, 22, 4, 4, Yes], [22, 23, 3, 3, Yes], [23, 0, 1, 1, Yes], [23, 1, 0, 0, Yes], [23, 2, 23, 23, Yes], [23, 3, 22, 22, Yes], [23, 4, 21, 21, Yes], [23, 5, 20, 20, Yes], [23, 6, 19, 19, Yes], [23, 7, 18, 18, Yes], [23, 8, 17, 17, Yes], [23, 9, 16, 16, Yes], [23, 10, 15, 15, Yes], [23, 11, 14, 14, Yes], [23, 12, 13, 13, Yes], [23, 13, 12, 12, Yes], [23, 14, 11, 11, Yes], [23, 15, 10, 10, Yes], [23, 16, 9, 9, Yes], [23, 17, 8, 8, Yes], [23, 18, 7, 7, Yes], [23, 19, 6, 6, Yes], [23, 20, 5, 5, Yes], [23, 21, 4, 4, Yes], [23, 22, 3, 3, Yes], [23, 23, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (24, 24, [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 23, 23, Yes], [0, 2, 22, 22, Yes], [0, 3, 21, 21, Yes], [0, 4, 20, 20, Yes], [0, 5, 19, 19, Yes], [0, 6, 18, 18, Yes], [0, 7, 17, 17, Yes], [0, 8, 16, 16, Yes], [0, 9, 15, 15, Yes], [0, 10, 14, 14, Yes], [0, 11, 13, 13, Yes], [0, 12, 12, 12, Yes], [0, 13, 11, 11, Yes], [0, 14, 10, 10, Yes], [0, 15, 9, 9, Yes], [0, 16, 8, 8, Yes], [0, 17, 7, 7, Yes], [0, 18, 6, 6, Yes], [0, 19, 5, 5, Yes], [0, 20, 4, 4, Yes], [0, 21, 3, 3, Yes], [0, 22, 2, 2, Yes], [0, 23, 1, 1, Yes], [1, 0, 23, 23, Yes], [1, 1, 22, 22, Yes], [1, 2, 21, 21, Yes], [1, 3, 20, 20, Yes], [1, 4, 19, 19, Yes], [1, 5, 18, 18, Yes], [1, 6, 17, 17, Yes], [1, 7, 16, 16, Yes], [1, 8, 15, 15, Yes], [1, 9, 14, 14, Yes], [1, 10, 13, 13, Yes], [1, 11, 12, 12, Yes], [1, 12, 11, 11, Yes], [1, 13, 10, 10, Yes], [1, 14, 9, 9, Yes], [1, 15, 8, 8, Yes], [1, 16, 7, 7, Yes], [1, 17, 6, 6, Yes], [1, 18, 5, 5, Yes], [1, 19, 4, 4, Yes], [1, 20, 3, 3, Yes], [1, 21, 2, 2, Yes], [1, 22, 1, 1, Yes], [1, 23, 0, 0, Yes], [2, 0, 22, 22, Yes], [2, 1, 21, 21, Yes], [2, 2, 20, 20, Yes], [2, 3, 19, 19, Yes], [2, 4, 18, 18, Yes], [2, 5, 17, 17, Yes], [2, 6, 16, 16, Yes], [2, 7, 15, 15, Yes], [2, 8, 14, 14, Yes], [2, 9, 13, 13, Yes], [2, 10, 12, 12, Yes], [2, 11, 11, 11, Yes], [2, 12, 10, 10, Yes], [2, 13, 9, 9, Yes], [2, 14, 8, 8, Yes], [2, 15, 7, 7, Yes], [2, 16, 6, 6, Yes], [2, 17, 5, 5, Yes], [2, 18, 4, 4, Yes], [2, 19, 3, 3, Yes], [2, 20, 2, 2, Yes], [2, 21, 1, 1, Yes], [2, 22, 0, 0, Yes], [2, 23, 23, 23, Yes], [3, 0, 21, 21, Yes], [3, 1, 20, 20, Yes], [3, 2, 19, 19, Yes], [3, 3, 18, 18, Yes], [3, 4, 17, 17, Yes], [3, 5, 16, 16, Yes], [3, 6, 15, 15, Yes], [3, 7, 14, 14, Yes], [3, 8, 13, 13, Yes], [3, 9, 12, 12, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [3, 12, 9, 9, Yes], [3, 13, 8, 8, Yes], [3, 14, 7, 7, Yes], [3, 15, 6, 6, Yes], [3, 16, 5, 5, Yes], [3, 17, 4, 4, Yes], [3, 18, 3, 3, Yes], [3, 19, 2, 2, Yes], [3, 20, 1, 1, Yes], [3, 21, 0, 0, Yes], [3, 22, 23, 23, Yes], [3, 23, 22, 22, Yes], [4, 0, 20, 20, Yes], [4, 1, 19, 19, Yes], [4, 2, 18, 18, Yes], [4, 3, 17, 17, Yes], [4, 4, 16, 16, Yes], [4, 5, 15, 15, Yes], [4, 6, 14, 14, Yes], [4, 7, 13, 13, Yes], [4, 8, 12, 12, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [4, 12, 8, 8, Yes], [4, 13, 7, 7, Yes], [4, 14, 6, 6, Yes], [4, 15, 5, 5, Yes], [4, 16, 4, 4, Yes], [4, 17, 3, 3, Yes], [4, 18, 2, 2, Yes], [4, 19, 1, 1, Yes], [4, 20, 0, 0, Yes], [4, 21, 23, 23, Yes], [4, 22, 22, 22, Yes], [4, 23, 21, 21, Yes], [5, 0, 19, 19, Yes], [5, 1, 18, 18, Yes], [5, 2, 17, 17, Yes], [5, 3, 16, 16, Yes], [5, 4, 15, 15, Yes], [5, 5, 14, 14, Yes], [5, 6, 13, 13, Yes], [5, 7, 12, 12, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [5, 12, 7, 7, Yes], [5, 13, 6, 6, Yes], [5, 14, 5, 5, Yes], [5, 15, 4, 4, Yes], [5, 16, 3, 3, Yes], [5, 17, 2, 2, Yes], [5, 18, 1, 1, Yes], [5, 19, 0, 0, Yes], [5, 20, 23, 23, Yes], [5, 21, 22, 22, Yes], [5, 22, 21, 21, Yes], [5, 23, 20, 20, Yes], [6, 0, 18, 18, Yes], [6, 1, 17, 17, Yes], [6, 2, 16, 16, Yes], [6, 3, 15, 15, Yes], [6, 4, 14, 14, Yes], [6, 5, 13, 13, Yes], [6, 6, 12, 12, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [6, 12, 6, 6, Yes], [6, 13, 5, 5, Yes], [6, 14, 4, 4, Yes], [6, 15, 3, 3, Yes], [6, 16, 2, 2, Yes], [6, 17, 1, 1, Yes], [6, 18, 0, 0, Yes], [6, 19, 23, 23, Yes], [6, 20, 22, 22, Yes], [6, 21, 21, 21, Yes], [6, 22, 20, 20, Yes], [6, 23, 19, 19, Yes], [7, 0, 17, 17, Yes], [7, 1, 16, 16, Yes], [7, 2, 15, 15, Yes], [7, 3, 14, 14, Yes], [7, 4, 13, 13, Yes], [7, 5, 12, 12, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [7, 12, 5, 5, Yes], [7, 13, 4, 4, Yes], [7, 14, 3, 3, Yes], [7, 15, 2, 2, Yes], [7, 16, 1, 1, Yes], [7, 17, 0, 0, Yes], [7, 18, 23, 23, Yes], [7, 19, 22, 22, Yes], [7, 20, 21, 21, Yes], [7, 21, 20, 20, Yes], [7, 22, 19, 19, Yes], [7, 23, 18, 18, Yes], [8, 0, 16, 16, Yes], [8, 1, 15, 15, Yes], [8, 2, 14, 14, Yes], [8, 3, 13, 13, Yes], [8, 4, 12, 12, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [8, 12, 4, 4, Yes], [8, 13, 3, 3, Yes], [8, 14, 2, 2, Yes], [8, 15, 1, 1, Yes], [8, 16, 0, 0, Yes], [8, 17, 23, 23, Yes], [8, 18, 22, 22, Yes], [8, 19, 21, 21, Yes], [8, 20, 20, 20, Yes], [8, 21, 19, 19, Yes], [8, 22, 18, 18, Yes], [8, 23, 17, 17, Yes], [9, 0, 15, 15, Yes], [9, 1, 14, 14, Yes], [9, 2, 13, 13, Yes], [9, 3, 12, 12, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [9, 12, 3, 3, Yes], [9, 13, 2, 2, Yes], [9, 14, 1, 1, Yes], [9, 15, 0, 0, Yes], [9, 16, 23, 23, Yes], [9, 17, 22, 22, Yes], [9, 18, 21, 21, Yes], [9, 19, 20, 20, Yes], [9, 20, 19, 19, Yes], [9, 21, 18, 18, Yes], [9, 22, 17, 17, Yes], [9, 23, 16, 16, Yes], [10, 0, 14, 14, Yes], [10, 1, 13, 13, Yes], [10, 2, 12, 12, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [10, 12, 2, 2, Yes], [10, 13, 1, 1, Yes], [10, 14, 0, 0, Yes], [10, 15, 23, 23, Yes], [10, 16, 22, 22, Yes], [10, 17, 21, 21, Yes], [10, 18, 20, 20, Yes], [10, 19, 19, 19, Yes], [10, 20, 18, 18, Yes], [10, 21, 17, 17, Yes], [10, 22, 16, 16, Yes], [10, 23, 15, 15, Yes], [11, 0, 13, 13, Yes], [11, 1, 12, 12, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes], [11, 12, 1, 1, Yes], [11, 13, 0, 0, Yes], [11, 14, 23, 23, Yes], [11, 15, 22, 22, Yes], [11, 16, 21, 21, Yes], [11, 17, 20, 20, Yes], [11, 18, 19, 19, Yes], [11, 19, 18, 18, Yes], [11, 20, 17, 17, Yes], [11, 21, 16, 16, Yes], [11, 22, 15, 15, Yes], [11, 23, 14, 14, Yes], [12, 0, 12, 12, Yes], [12, 1, 11, 11, Yes], [12, 2, 10, 10, Yes], [12, 3, 9, 9, Yes], [12, 4, 8, 8, Yes], [12, 5, 7, 7, Yes], [12, 6, 6, 6, Yes], [12, 7, 5, 5, Yes], [12, 8, 4, 4, Yes], [12, 9, 3, 3, Yes], [12, 10, 2, 2, Yes], [12, 11, 1, 1, Yes], [12, 12, 0, 0, Yes], [12, 13, 23, 23, Yes], [12, 14, 22, 22, Yes], [12, 15, 21, 21, Yes], [12, 16, 20, 20, Yes], [12, 17, 19, 19, Yes], [12, 18, 18, 18, Yes], [12, 19, 17, 17, Yes], [12, 20, 16, 16, Yes], [12, 21, 15, 15, Yes], [12, 22, 14, 14, Yes], [12, 23, 13, 13, Yes], [13, 0, 11, 11, Yes], [13, 1, 10, 10, Yes], [13, 2, 9, 9, Yes], [13, 3, 8, 8, Yes], [13, 4, 7, 7, Yes], [13, 5, 6, 6, Yes], [13, 6, 5, 5, Yes], [13, 7, 4, 4, Yes], [13, 8, 3, 3, Yes], [13, 9, 2, 2, Yes], [13, 10, 1, 1, Yes], [13, 11, 0, 0, Yes], [13, 12, 23, 23, Yes], [13, 13, 22, 22, Yes], [13, 14, 21, 21, Yes], [13, 15, 20, 20, Yes], [13, 16, 19, 19, Yes], [13, 17, 18, 18, Yes], [13, 18, 17, 17, Yes], [13, 19, 16, 16, Yes], [13, 20, 15, 15, Yes], [13, 21, 14, 14, Yes], [13, 22, 13, 13, Yes], [13, 23, 12, 12, Yes], [14, 0, 10, 10, Yes], [14, 1, 9, 9, Yes], [14, 2, 8, 8, Yes], [14, 3, 7, 7, Yes], [14, 4, 6, 6, Yes], [14, 5, 5, 5, Yes], [14, 6, 4, 4, Yes], [14, 7, 3, 3, Yes], [14, 8, 2, 2, Yes], [14, 9, 1, 1, Yes], [14, 10, 0, 0, Yes], [14, 11, 23, 23, Yes], [14, 12, 22, 22, Yes], [14, 13, 21, 21, Yes], [14, 14, 20, 20, Yes], [14, 15, 19, 19, Yes], [14, 16, 18, 18, Yes], [14, 17, 17, 17, Yes], [14, 18, 16, 16, Yes], [14, 19, 15, 15, Yes], [14, 20, 14, 14, Yes], [14, 21, 13, 13, Yes], [14, 22, 12, 12, Yes], [14, 23, 11, 11, Yes], [15, 0, 9, 9, Yes], [15, 1, 8, 8, Yes], [15, 2, 7, 7, Yes], [15, 3, 6, 6, Yes], [15, 4, 5, 5, Yes], [15, 5, 4, 4, Yes], [15, 6, 3, 3, Yes], [15, 7, 2, 2, Yes], [15, 8, 1, 1, Yes], [15, 9, 0, 0, Yes], [15, 10, 23, 23, Yes], [15, 11, 22, 22, Yes], [15, 12, 21, 21, Yes], [15, 13, 20, 20, Yes], [15, 14, 19, 19, Yes], [15, 15, 18, 18, Yes], [15, 16, 17, 17, Yes], [15, 17, 16, 16, Yes], [15, 18, 15, 15, Yes], [15, 19, 14, 14, Yes], [15, 20, 13, 13, Yes], [15, 21, 12, 12, Yes], [15, 22, 11, 11, Yes], [15, 23, 10, 10, Yes], [16, 0, 8, 8, Yes], [16, 1, 7, 7, Yes], [16, 2, 6, 6, Yes], [16, 3, 5, 5, Yes], [16, 4, 4, 4, Yes], [16, 5, 3, 3, Yes], [16, 6, 2, 2, Yes], [16, 7, 1, 1, Yes], [16, 8, 0, 0, Yes], [16, 9, 23, 23, Yes], [16, 10, 22, 22, Yes], [16, 11, 21, 21, Yes], [16, 12, 20, 20, Yes], [16, 13, 19, 19, Yes], [16, 14, 18, 18, Yes], [16, 15, 17, 17, Yes], [16, 16, 16, 16, Yes], [16, 17, 15, 15, Yes], [16, 18, 14, 14, Yes], [16, 19, 13, 13, Yes], [16, 20, 12, 12, Yes], [16, 21, 11, 11, Yes], [16, 22, 10, 10, Yes], [16, 23, 9, 9, Yes], [17, 0, 7, 7, Yes], [17, 1, 6, 6, Yes], [17, 2, 5, 5, Yes], [17, 3, 4, 4, Yes], [17, 4, 3, 3, Yes], [17, 5, 2, 2, Yes], [17, 6, 1, 1, Yes], [17, 7, 0, 0, Yes], [17, 8, 23, 23, Yes], [17, 9, 22, 22, Yes], [17, 10, 21, 21, Yes], [17, 11, 20, 20, Yes], [17, 12, 19, 19, Yes], [17, 13, 18, 18, Yes], [17, 14, 17, 17, Yes], [17, 15, 16, 16, Yes], [17, 16, 15, 15, Yes], [17, 17, 14, 14, Yes], [17, 18, 13, 13, Yes], [17, 19, 12, 12, Yes], [17, 20, 11, 11, Yes], [17, 21, 10, 10, Yes], [17, 22, 9, 9, Yes], [17, 23, 8, 8, Yes], [18, 0, 6, 6, Yes], [18, 1, 5, 5, Yes], [18, 2, 4, 4, Yes], [18, 3, 3, 3, Yes], [18, 4, 2, 2, Yes], [18, 5, 1, 1, Yes], [18, 6, 0, 0, Yes], [18, 7, 23, 23, Yes], [18, 8, 22, 22, Yes], [18, 9, 21, 21, Yes], [18, 10, 20, 20, Yes], [18, 11, 19, 19, Yes], [18, 12, 18, 18, Yes], [18, 13, 17, 17, Yes], [18, 14, 16, 16, Yes], [18, 15, 15, 15, Yes], [18, 16, 14, 14, Yes], [18, 17, 13, 13, Yes], [18, 18, 12, 12, Yes], [18, 19, 11, 11, Yes], [18, 20, 10, 10, Yes], [18, 21, 9, 9, Yes], [18, 22, 8, 8, Yes], [18, 23, 7, 7, Yes], [19, 0, 5, 5, Yes], [19, 1, 4, 4, Yes], [19, 2, 3, 3, Yes], [19, 3, 2, 2, Yes], [19, 4, 1, 1, Yes], [19, 5, 0, 0, Yes], [19, 6, 23, 23, Yes], [19, 7, 22, 22, Yes], [19, 8, 21, 21, Yes], [19, 9, 20, 20, Yes], [19, 10, 19, 19, Yes], [19, 11, 18, 18, Yes], [19, 12, 17, 17, Yes], [19, 13, 16, 16, Yes], [19, 14, 15, 15, Yes], [19, 15, 14, 14, Yes], [19, 16, 13, 13, Yes], [19, 17, 12, 12, Yes], [19, 18, 11, 11, Yes], [19, 19, 10, 10, Yes], [19, 20, 9, 9, Yes], [19, 21, 8, 8, Yes], [19, 22, 7, 7, Yes], [19, 23, 6, 6, Yes], [20, 0, 4, 4, Yes], [20, 1, 3, 3, Yes], [20, 2, 2, 2, Yes], [20, 3, 1, 1, Yes], [20, 4, 0, 0, Yes], [20, 5, 23, 23, Yes], [20, 6, 22, 22, Yes], [20, 7, 21, 21, Yes], [20, 8, 20, 20, Yes], [20, 9, 19, 19, Yes], [20, 10, 18, 18, Yes], [20, 11, 17, 17, Yes], [20, 12, 16, 16, Yes], [20, 13, 15, 15, Yes], [20, 14, 14, 14, Yes], [20, 15, 13, 13, Yes], [20, 16, 12, 12, Yes], [20, 17, 11, 11, Yes], [20, 18, 10, 10, Yes], [20, 19, 9, 9, Yes], [20, 20, 8, 8, Yes], [20, 21, 7, 7, Yes], [20, 22, 6, 6, Yes], [20, 23, 5, 5, Yes], [21, 0, 3, 3, Yes], [21, 1, 2, 2, Yes], [21, 2, 1, 1, Yes], [21, 3, 0, 0, Yes], [21, 4, 23, 23, Yes], [21, 5, 22, 22, Yes], [21, 6, 21, 21, Yes], [21, 7, 20, 20, Yes], [21, 8, 19, 19, Yes], [21, 9, 18, 18, Yes], [21, 10, 17, 17, Yes], [21, 11, 16, 16, Yes], [21, 12, 15, 15, Yes], [21, 13, 14, 14, Yes], [21, 14, 13, 13, Yes], [21, 15, 12, 12, Yes], [21, 16, 11, 11, Yes], [21, 17, 10, 10, Yes], [21, 18, 9, 9, Yes], [21, 19, 8, 8, Yes], [21, 20, 7, 7, Yes], [21, 21, 6, 6, Yes], [21, 22, 5, 5, Yes], [21, 23, 4, 4, Yes], [22, 0, 2, 2, Yes], [22, 1, 1, 1, Yes], [22, 2, 0, 0, Yes], [22, 3, 23, 23, Yes], [22, 4, 22, 22, Yes], [22, 5, 21, 21, Yes], [22, 6, 20, 20, Yes], [22, 7, 19, 19, Yes], [22, 8, 18, 18, Yes], [22, 9, 17, 17, Yes], [22, 10, 16, 16, Yes], [22, 11, 15, 15, Yes], [22, 12, 14, 14, Yes], [22, 13, 13, 13, Yes], [22, 14, 12, 12, Yes], [22, 15, 11, 11, Yes], [22, 16, 10, 10, Yes], [22, 17, 9, 9, Yes], [22, 18, 8, 8, Yes], [22, 19, 7, 7, Yes], [22, 20, 6, 6, Yes], [22, 21, 5, 5, Yes], [22, 22, 4, 4, Yes], [22, 23, 3, 3, Yes], [23, 0, 1, 1, Yes], [23, 1, 0, 0, Yes], [23, 2, 23, 23, Yes], [23, 3, 22, 22, Yes], [23, 4, 21, 21, Yes], [23, 5, 20, 20, Yes], [23, 6, 19, 19, Yes], [23, 7, 18, 18, Yes], [23, 8, 17, 17, Yes], [23, 9, 16, 16, Yes], [23, 10, 15, 15, Yes], [23, 11, 14, 14, Yes], [23, 12, 13, 13, Yes], [23, 13, 12, 12, Yes], [23, 14, 11, 11, Yes], [23, 15, 10, 10, Yes], [23, 16, 9, 9, Yes], [23, 17, 8, 8, Yes], [23, 18, 7, 7, Yes], [23, 19, 6, 6, Yes], [23, 20, 5, 5, Yes], [23, 21, 4, 4, Yes], [23, 22, 3, 3, Yes], [23, 23, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0023","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47130400fbc19f578140c88fd7eb3797f060a772161cc86da357eac385ff5ce9","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141023,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], target identity=0, and total mapping=[0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(25, 25, [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 24, 24, Yes], [0, 2, 23, 23, Yes], [0, 3, 22, 22, Yes], [0, 4, 21, 21, Yes], [0, 5, 20, 20, Yes], [0, 6, 19, 19, Yes], [0, 7, 18, 18, Yes], [0, 8, 17, 17, Yes], [0, 9, 16, 16, Yes], [0, 10, 15, 15, Yes], [0, 11, 14, 14, Yes], [0, 12, 13, 13, Yes], [0, 13, 12, 12, Yes], [0, 14, 11, 11, Yes], [0, 15, 10, 10, Yes], [0, 16, 9, 9, Yes], [0, 17, 8, 8, Yes], [0, 18, 7, 7, Yes], [0, 19, 6, 6, Yes], [0, 20, 5, 5, Yes], [0, 21, 4, 4, Yes], [0, 22, 3, 3, Yes], [0, 23, 2, 2, Yes], [0, 24, 1, 1, Yes], [1, 0, 24, 24, Yes], [1, 1, 23, 23, Yes], [1, 2, 22, 22, Yes], [1, 3, 21, 21, Yes], [1, 4, 20, 20, Yes], [1, 5, 19, 19, Yes], [1, 6, 18, 18, Yes], [1, 7, 17, 17, Yes], [1, 8, 16, 16, Yes], [1, 9, 15, 15, Yes], [1, 10, 14, 14, Yes], [1, 11, 13, 13, Yes], [1, 12, 12, 12, Yes], [1, 13, 11, 11, Yes], [1, 14, 10, 10, Yes], [1, 15, 9, 9, Yes], [1, 16, 8, 8, Yes], [1, 17, 7, 7, Yes], [1, 18, 6, 6, Yes], [1, 19, 5, 5, Yes], [1, 20, 4, 4, Yes], [1, 21, 3, 3, Yes], [1, 22, 2, 2, Yes], [1, 23, 1, 1, Yes], [1, 24, 0, 0, Yes], [2, 0, 23, 23, Yes], [2, 1, 22, 22, Yes], [2, 2, 21, 21, Yes], [2, 3, 20, 20, Yes], [2, 4, 19, 19, Yes], [2, 5, 18, 18, Yes], [2, 6, 17, 17, Yes], [2, 7, 16, 16, Yes], [2, 8, 15, 15, Yes], [2, 9, 14, 14, Yes], [2, 10, 13, 13, Yes], [2, 11, 12, 12, Yes], [2, 12, 11, 11, Yes], [2, 13, 10, 10, Yes], [2, 14, 9, 9, Yes], [2, 15, 8, 8, Yes], [2, 16, 7, 7, Yes], [2, 17, 6, 6, Yes], [2, 18, 5, 5, Yes], [2, 19, 4, 4, Yes], [2, 20, 3, 3, Yes], [2, 21, 2, 2, Yes], [2, 22, 1, 1, Yes], [2, 23, 0, 0, Yes], [2, 24, 24, 24, Yes], [3, 0, 22, 22, Yes], [3, 1, 21, 21, Yes], [3, 2, 20, 20, Yes], [3, 3, 19, 19, Yes], [3, 4, 18, 18, Yes], [3, 5, 17, 17, Yes], [3, 6, 16, 16, Yes], [3, 7, 15, 15, Yes], [3, 8, 14, 14, Yes], [3, 9, 13, 13, Yes], [3, 10, 12, 12, Yes], [3, 11, 11, 11, Yes], [3, 12, 10, 10, Yes], [3, 13, 9, 9, Yes], [3, 14, 8, 8, Yes], [3, 15, 7, 7, Yes], [3, 16, 6, 6, Yes], [3, 17, 5, 5, Yes], [3, 18, 4, 4, Yes], [3, 19, 3, 3, Yes], [3, 20, 2, 2, Yes], [3, 21, 1, 1, Yes], [3, 22, 0, 0, Yes], [3, 23, 24, 24, Yes], [3, 24, 23, 23, Yes], [4, 0, 21, 21, Yes], [4, 1, 20, 20, Yes], [4, 2, 19, 19, Yes], [4, 3, 18, 18, Yes], [4, 4, 17, 17, Yes], [4, 5, 16, 16, Yes], [4, 6, 15, 15, Yes], [4, 7, 14, 14, Yes], [4, 8, 13, 13, Yes], [4, 9, 12, 12, Yes], [4, 10, 11, 11, Yes], [4, 11, 10, 10, Yes], [4, 12, 9, 9, Yes], [4, 13, 8, 8, Yes], [4, 14, 7, 7, Yes], [4, 15, 6, 6, Yes], [4, 16, 5, 5, Yes], [4, 17, 4, 4, Yes], [4, 18, 3, 3, Yes], [4, 19, 2, 2, Yes], [4, 20, 1, 1, Yes], [4, 21, 0, 0, Yes], [4, 22, 24, 24, Yes], [4, 23, 23, 23, Yes], [4, 24, 22, 22, Yes], [5, 0, 20, 20, Yes], [5, 1, 19, 19, Yes], [5, 2, 18, 18, Yes], [5, 3, 17, 17, Yes], [5, 4, 16, 16, Yes], [5, 5, 15, 15, Yes], [5, 6, 14, 14, Yes], [5, 7, 13, 13, Yes], [5, 8, 12, 12, Yes], [5, 9, 11, 11, Yes], [5, 10, 10, 10, Yes], [5, 11, 9, 9, Yes], [5, 12, 8, 8, Yes], [5, 13, 7, 7, Yes], [5, 14, 6, 6, Yes], [5, 15, 5, 5, Yes], [5, 16, 4, 4, Yes], [5, 17, 3, 3, Yes], [5, 18, 2, 2, Yes], [5, 19, 1, 1, Yes], [5, 20, 0, 0, Yes], [5, 21, 24, 24, Yes], [5, 22, 23, 23, Yes], [5, 23, 22, 22, Yes], [5, 24, 21, 21, Yes], [6, 0, 19, 19, Yes], [6, 1, 18, 18, Yes], [6, 2, 17, 17, Yes], [6, 3, 16, 16, Yes], [6, 4, 15, 15, Yes], [6, 5, 14, 14, Yes], [6, 6, 13, 13, Yes], [6, 7, 12, 12, Yes], [6, 8, 11, 11, Yes], [6, 9, 10, 10, Yes], [6, 10, 9, 9, Yes], [6, 11, 8, 8, Yes], [6, 12, 7, 7, Yes], [6, 13, 6, 6, Yes], [6, 14, 5, 5, Yes], [6, 15, 4, 4, Yes], [6, 16, 3, 3, Yes], [6, 17, 2, 2, Yes], [6, 18, 1, 1, Yes], [6, 19, 0, 0, Yes], [6, 20, 24, 24, Yes], [6, 21, 23, 23, Yes], [6, 22, 22, 22, Yes], [6, 23, 21, 21, Yes], [6, 24, 20, 20, Yes], [7, 0, 18, 18, Yes], [7, 1, 17, 17, Yes], [7, 2, 16, 16, Yes], [7, 3, 15, 15, Yes], [7, 4, 14, 14, Yes], [7, 5, 13, 13, Yes], [7, 6, 12, 12, Yes], [7, 7, 11, 11, Yes], [7, 8, 10, 10, Yes], [7, 9, 9, 9, Yes], [7, 10, 8, 8, Yes], [7, 11, 7, 7, Yes], [7, 12, 6, 6, Yes], [7, 13, 5, 5, Yes], [7, 14, 4, 4, Yes], [7, 15, 3, 3, Yes], [7, 16, 2, 2, Yes], [7, 17, 1, 1, Yes], [7, 18, 0, 0, Yes], [7, 19, 24, 24, Yes], [7, 20, 23, 23, Yes], [7, 21, 22, 22, Yes], [7, 22, 21, 21, Yes], [7, 23, 20, 20, Yes], [7, 24, 19, 19, Yes], [8, 0, 17, 17, Yes], [8, 1, 16, 16, Yes], [8, 2, 15, 15, Yes], [8, 3, 14, 14, Yes], [8, 4, 13, 13, Yes], [8, 5, 12, 12, Yes], [8, 6, 11, 11, Yes], [8, 7, 10, 10, Yes], [8, 8, 9, 9, Yes], [8, 9, 8, 8, Yes], [8, 10, 7, 7, Yes], [8, 11, 6, 6, Yes], [8, 12, 5, 5, Yes], [8, 13, 4, 4, Yes], [8, 14, 3, 3, Yes], [8, 15, 2, 2, Yes], [8, 16, 1, 1, Yes], [8, 17, 0, 0, Yes], [8, 18, 24, 24, Yes], [8, 19, 23, 23, Yes], [8, 20, 22, 22, Yes], [8, 21, 21, 21, Yes], [8, 22, 20, 20, Yes], [8, 23, 19, 19, Yes], [8, 24, 18, 18, Yes], [9, 0, 16, 16, Yes], [9, 1, 15, 15, Yes], [9, 2, 14, 14, Yes], [9, 3, 13, 13, Yes], [9, 4, 12, 12, Yes], [9, 5, 11, 11, Yes], [9, 6, 10, 10, Yes], [9, 7, 9, 9, Yes], [9, 8, 8, 8, Yes], [9, 9, 7, 7, Yes], [9, 10, 6, 6, Yes], [9, 11, 5, 5, Yes], [9, 12, 4, 4, Yes], [9, 13, 3, 3, Yes], [9, 14, 2, 2, Yes], [9, 15, 1, 1, Yes], [9, 16, 0, 0, Yes], [9, 17, 24, 24, Yes], [9, 18, 23, 23, Yes], [9, 19, 22, 22, Yes], [9, 20, 21, 21, Yes], [9, 21, 20, 20, Yes], [9, 22, 19, 19, Yes], [9, 23, 18, 18, Yes], [9, 24, 17, 17, Yes], [10, 0, 15, 15, Yes], [10, 1, 14, 14, Yes], [10, 2, 13, 13, Yes], [10, 3, 12, 12, Yes], [10, 4, 11, 11, Yes], [10, 5, 10, 10, Yes], [10, 6, 9, 9, Yes], [10, 7, 8, 8, Yes], [10, 8, 7, 7, Yes], [10, 9, 6, 6, Yes], [10, 10, 5, 5, Yes], [10, 11, 4, 4, Yes], [10, 12, 3, 3, Yes], [10, 13, 2, 2, Yes], [10, 14, 1, 1, Yes], [10, 15, 0, 0, Yes], [10, 16, 24, 24, Yes], [10, 17, 23, 23, Yes], [10, 18, 22, 22, Yes], [10, 19, 21, 21, Yes], [10, 20, 20, 20, Yes], [10, 21, 19, 19, Yes], [10, 22, 18, 18, Yes], [10, 23, 17, 17, Yes], [10, 24, 16, 16, Yes], [11, 0, 14, 14, Yes], [11, 1, 13, 13, Yes], [11, 2, 12, 12, Yes], [11, 3, 11, 11, Yes], [11, 4, 10, 10, Yes], [11, 5, 9, 9, Yes], [11, 6, 8, 8, Yes], [11, 7, 7, 7, Yes], [11, 8, 6, 6, Yes], [11, 9, 5, 5, Yes], [11, 10, 4, 4, Yes], [11, 11, 3, 3, Yes], [11, 12, 2, 2, Yes], [11, 13, 1, 1, Yes], [11, 14, 0, 0, Yes], [11, 15, 24, 24, Yes], [11, 16, 23, 23, Yes], [11, 17, 22, 22, Yes], [11, 18, 21, 21, Yes], [11, 19, 20, 20, Yes], [11, 20, 19, 19, Yes], [11, 21, 18, 18, Yes], [11, 22, 17, 17, Yes], [11, 23, 16, 16, Yes], [11, 24, 15, 15, Yes], [12, 0, 13, 13, Yes], [12, 1, 12, 12, Yes], [12, 2, 11, 11, Yes], [12, 3, 10, 10, Yes], [12, 4, 9, 9, Yes], [12, 5, 8, 8, Yes], [12, 6, 7, 7, Yes], [12, 7, 6, 6, Yes], [12, 8, 5, 5, Yes], [12, 9, 4, 4, Yes], [12, 10, 3, 3, Yes], [12, 11, 2, 2, Yes], [12, 12, 1, 1, Yes], [12, 13, 0, 0, Yes], [12, 14, 24, 24, Yes], [12, 15, 23, 23, Yes], [12, 16, 22, 22, Yes], [12, 17, 21, 21, Yes], [12, 18, 20, 20, Yes], [12, 19, 19, 19, Yes], [12, 20, 18, 18, Yes], [12, 21, 17, 17, Yes], [12, 22, 16, 16, Yes], [12, 23, 15, 15, Yes], [12, 24, 14, 14, Yes], [13, 0, 12, 12, Yes], [13, 1, 11, 11, Yes], [13, 2, 10, 10, Yes], [13, 3, 9, 9, Yes], [13, 4, 8, 8, Yes], [13, 5, 7, 7, Yes], [13, 6, 6, 6, Yes], [13, 7, 5, 5, Yes], [13, 8, 4, 4, Yes], [13, 9, 3, 3, Yes], [13, 10, 2, 2, Yes], [13, 11, 1, 1, Yes], [13, 12, 0, 0, Yes], [13, 13, 24, 24, Yes], [13, 14, 23, 23, Yes], [13, 15, 22, 22, Yes], [13, 16, 21, 21, Yes], [13, 17, 20, 20, Yes], [13, 18, 19, 19, Yes], [13, 19, 18, 18, Yes], [13, 20, 17, 17, Yes], [13, 21, 16, 16, Yes], [13, 22, 15, 15, Yes], [13, 23, 14, 14, Yes], [13, 24, 13, 13, Yes], [14, 0, 11, 11, Yes], [14, 1, 10, 10, Yes], [14, 2, 9, 9, Yes], [14, 3, 8, 8, Yes], [14, 4, 7, 7, Yes], [14, 5, 6, 6, Yes], [14, 6, 5, 5, Yes], [14, 7, 4, 4, Yes], [14, 8, 3, 3, Yes], [14, 9, 2, 2, Yes], [14, 10, 1, 1, Yes], [14, 11, 0, 0, Yes], [14, 12, 24, 24, Yes], [14, 13, 23, 23, Yes], [14, 14, 22, 22, Yes], [14, 15, 21, 21, Yes], [14, 16, 20, 20, Yes], [14, 17, 19, 19, Yes], [14, 18, 18, 18, Yes], [14, 19, 17, 17, Yes], [14, 20, 16, 16, Yes], [14, 21, 15, 15, Yes], [14, 22, 14, 14, Yes], [14, 23, 13, 13, Yes], [14, 24, 12, 12, Yes], [15, 0, 10, 10, Yes], [15, 1, 9, 9, Yes], [15, 2, 8, 8, Yes], [15, 3, 7, 7, Yes], [15, 4, 6, 6, Yes], [15, 5, 5, 5, Yes], [15, 6, 4, 4, Yes], [15, 7, 3, 3, Yes], [15, 8, 2, 2, Yes], [15, 9, 1, 1, Yes], [15, 10, 0, 0, Yes], [15, 11, 24, 24, Yes], [15, 12, 23, 23, Yes], [15, 13, 22, 22, Yes], [15, 14, 21, 21, Yes], [15, 15, 20, 20, Yes], [15, 16, 19, 19, Yes], [15, 17, 18, 18, Yes], [15, 18, 17, 17, Yes], [15, 19, 16, 16, Yes], [15, 20, 15, 15, Yes], [15, 21, 14, 14, Yes], [15, 22, 13, 13, Yes], [15, 23, 12, 12, Yes], [15, 24, 11, 11, Yes], [16, 0, 9, 9, Yes], [16, 1, 8, 8, Yes], [16, 2, 7, 7, Yes], [16, 3, 6, 6, Yes], [16, 4, 5, 5, Yes], [16, 5, 4, 4, Yes], [16, 6, 3, 3, Yes], [16, 7, 2, 2, Yes], [16, 8, 1, 1, Yes], [16, 9, 0, 0, Yes], [16, 10, 24, 24, Yes], [16, 11, 23, 23, Yes], [16, 12, 22, 22, Yes], [16, 13, 21, 21, Yes], [16, 14, 20, 20, Yes], [16, 15, 19, 19, Yes], [16, 16, 18, 18, Yes], [16, 17, 17, 17, Yes], [16, 18, 16, 16, Yes], [16, 19, 15, 15, Yes], [16, 20, 14, 14, Yes], [16, 21, 13, 13, Yes], [16, 22, 12, 12, Yes], [16, 23, 11, 11, Yes], [16, 24, 10, 10, Yes], [17, 0, 8, 8, Yes], [17, 1, 7, 7, Yes], [17, 2, 6, 6, Yes], [17, 3, 5, 5, Yes], [17, 4, 4, 4, Yes], [17, 5, 3, 3, Yes], [17, 6, 2, 2, Yes], [17, 7, 1, 1, Yes], [17, 8, 0, 0, Yes], [17, 9, 24, 24, Yes], [17, 10, 23, 23, Yes], [17, 11, 22, 22, Yes], [17, 12, 21, 21, Yes], [17, 13, 20, 20, Yes], [17, 14, 19, 19, Yes], [17, 15, 18, 18, Yes], [17, 16, 17, 17, Yes], [17, 17, 16, 16, Yes], [17, 18, 15, 15, Yes], [17, 19, 14, 14, Yes], [17, 20, 13, 13, Yes], [17, 21, 12, 12, Yes], [17, 22, 11, 11, Yes], [17, 23, 10, 10, Yes], [17, 24, 9, 9, Yes], [18, 0, 7, 7, Yes], [18, 1, 6, 6, Yes], [18, 2, 5, 5, Yes], [18, 3, 4, 4, Yes], [18, 4, 3, 3, Yes], [18, 5, 2, 2, Yes], [18, 6, 1, 1, Yes], [18, 7, 0, 0, Yes], [18, 8, 24, 24, Yes], [18, 9, 23, 23, Yes], [18, 10, 22, 22, Yes], [18, 11, 21, 21, Yes], [18, 12, 20, 20, Yes], [18, 13, 19, 19, Yes], [18, 14, 18, 18, Yes], [18, 15, 17, 17, Yes], [18, 16, 16, 16, Yes], [18, 17, 15, 15, Yes], [18, 18, 14, 14, Yes], [18, 19, 13, 13, Yes], [18, 20, 12, 12, Yes], [18, 21, 11, 11, Yes], [18, 22, 10, 10, Yes], [18, 23, 9, 9, Yes], [18, 24, 8, 8, Yes], [19, 0, 6, 6, Yes], [19, 1, 5, 5, Yes], [19, 2, 4, 4, Yes], [19, 3, 3, 3, Yes], [19, 4, 2, 2, Yes], [19, 5, 1, 1, Yes], [19, 6, 0, 0, Yes], [19, 7, 24, 24, Yes], [19, 8, 23, 23, Yes], [19, 9, 22, 22, Yes], [19, 10, 21, 21, Yes], [19, 11, 20, 20, Yes], [19, 12, 19, 19, Yes], [19, 13, 18, 18, Yes], [19, 14, 17, 17, Yes], [19, 15, 16, 16, Yes], [19, 16, 15, 15, Yes], [19, 17, 14, 14, Yes], [19, 18, 13, 13, Yes], [19, 19, 12, 12, Yes], [19, 20, 11, 11, Yes], [19, 21, 10, 10, Yes], [19, 22, 9, 9, Yes], [19, 23, 8, 8, Yes], [19, 24, 7, 7, Yes], [20, 0, 5, 5, Yes], [20, 1, 4, 4, Yes], [20, 2, 3, 3, Yes], [20, 3, 2, 2, Yes], [20, 4, 1, 1, Yes], [20, 5, 0, 0, Yes], [20, 6, 24, 24, Yes], [20, 7, 23, 23, Yes], [20, 8, 22, 22, Yes], [20, 9, 21, 21, Yes], [20, 10, 20, 20, Yes], [20, 11, 19, 19, Yes], [20, 12, 18, 18, Yes], [20, 13, 17, 17, Yes], [20, 14, 16, 16, Yes], [20, 15, 15, 15, Yes], [20, 16, 14, 14, Yes], [20, 17, 13, 13, Yes], [20, 18, 12, 12, Yes], [20, 19, 11, 11, Yes], [20, 20, 10, 10, Yes], [20, 21, 9, 9, Yes], [20, 22, 8, 8, Yes], [20, 23, 7, 7, Yes], [20, 24, 6, 6, Yes], [21, 0, 4, 4, Yes], [21, 1, 3, 3, Yes], [21, 2, 2, 2, Yes], [21, 3, 1, 1, Yes], [21, 4, 0, 0, Yes], [21, 5, 24, 24, Yes], [21, 6, 23, 23, Yes], [21, 7, 22, 22, Yes], [21, 8, 21, 21, Yes], [21, 9, 20, 20, Yes], [21, 10, 19, 19, Yes], [21, 11, 18, 18, Yes], [21, 12, 17, 17, Yes], [21, 13, 16, 16, Yes], [21, 14, 15, 15, Yes], [21, 15, 14, 14, Yes], [21, 16, 13, 13, Yes], [21, 17, 12, 12, Yes], [21, 18, 11, 11, Yes], [21, 19, 10, 10, Yes], [21, 20, 9, 9, Yes], [21, 21, 8, 8, Yes], [21, 22, 7, 7, Yes], [21, 23, 6, 6, Yes], [21, 24, 5, 5, Yes], [22, 0, 3, 3, Yes], [22, 1, 2, 2, Yes], [22, 2, 1, 1, Yes], [22, 3, 0, 0, Yes], [22, 4, 24, 24, Yes], [22, 5, 23, 23, Yes], [22, 6, 22, 22, Yes], [22, 7, 21, 21, Yes], [22, 8, 20, 20, Yes], [22, 9, 19, 19, Yes], [22, 10, 18, 18, Yes], [22, 11, 17, 17, Yes], [22, 12, 16, 16, Yes], [22, 13, 15, 15, Yes], [22, 14, 14, 14, Yes], [22, 15, 13, 13, Yes], [22, 16, 12, 12, Yes], [22, 17, 11, 11, Yes], [22, 18, 10, 10, Yes], [22, 19, 9, 9, Yes], [22, 20, 8, 8, Yes], [22, 21, 7, 7, Yes], [22, 22, 6, 6, Yes], [22, 23, 5, 5, Yes], [22, 24, 4, 4, Yes], [23, 0, 2, 2, Yes], [23, 1, 1, 1, Yes], [23, 2, 0, 0, Yes], [23, 3, 24, 24, Yes], [23, 4, 23, 23, Yes], [23, 5, 22, 22, Yes], [23, 6, 21, 21, Yes], [23, 7, 20, 20, Yes], [23, 8, 19, 19, Yes], [23, 9, 18, 18, Yes], [23, 10, 17, 17, Yes], [23, 11, 16, 16, Yes], [23, 12, 15, 15, Yes], [23, 13, 14, 14, Yes], [23, 14, 13, 13, Yes], [23, 15, 12, 12, Yes], [23, 16, 11, 11, Yes], [23, 17, 10, 10, Yes], [23, 18, 9, 9, Yes], [23, 19, 8, 8, Yes], [23, 20, 7, 7, Yes], [23, 21, 6, 6, Yes], [23, 22, 5, 5, Yes], [23, 23, 4, 4, Yes], [23, 24, 3, 3, Yes], [24, 0, 1, 1, Yes], [24, 1, 0, 0, Yes], [24, 2, 24, 24, Yes], [24, 3, 23, 23, Yes], [24, 4, 22, 22, Yes], [24, 5, 21, 21, Yes], [24, 6, 20, 20, Yes], [24, 7, 19, 19, Yes], [24, 8, 18, 18, Yes], [24, 9, 17, 17, Yes], [24, 10, 16, 16, Yes], [24, 11, 15, 15, Yes], [24, 12, 14, 14, Yes], [24, 13, 13, 13, Yes], [24, 14, 12, 12, Yes], [24, 15, 11, 11, Yes], [24, 16, 10, 10, Yes], [24, 17, 9, 9, Yes], [24, 18, 8, 8, Yes], [24, 19, 7, 7, Yes], [24, 20, 6, 6, Yes], [24, 21, 5, 5, Yes], [24, 22, 4, 4, Yes], [24, 23, 3, 3, Yes], [24, 24, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (25, 25, [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 24, 24, Yes], [0, 2, 23, 23, Yes], [0, 3, 22, 22, Yes], [0, 4, 21, 21, Yes], [0, 5, 20, 20, Yes], [0, 6, 19, 19, Yes], [0, 7, 18, 18, Yes], [0, 8, 17, 17, Yes], [0, 9, 16, 16, Yes], [0, 10, 15, 15, Yes], [0, 11, 14, 14, Yes], [0, 12, 13, 13, Yes], [0, 13, 12, 12, Yes], [0, 14, 11, 11, Yes], [0, 15, 10, 10, Yes], [0, 16, 9, 9, Yes], [0, 17, 8, 8, Yes], [0, 18, 7, 7, Yes], [0, 19, 6, 6, Yes], [0, 20, 5, 5, Yes], [0, 21, 4, 4, Yes], [0, 22, 3, 3, Yes], [0, 23, 2, 2, Yes], [0, 24, 1, 1, Yes], [1, 0, 24, 24, Yes], [1, 1, 23, 23, Yes], [1, 2, 22, 22, Yes], [1, 3, 21, 21, Yes], [1, 4, 20, 20, Yes], [1, 5, 19, 19, Yes], [1, 6, 18, 18, Yes], [1, 7, 17, 17, Yes], [1, 8, 16, 16, Yes], [1, 9, 15, 15, Yes], [1, 10, 14, 14, Yes], [1, 11, 13, 13, Yes], [1, 12, 12, 12, Yes], [1, 13, 11, 11, Yes], [1, 14, 10, 10, Yes], [1, 15, 9, 9, Yes], [1, 16, 8, 8, Yes], [1, 17, 7, 7, Yes], [1, 18, 6, 6, Yes], [1, 19, 5, 5, Yes], [1, 20, 4, 4, Yes], [1, 21, 3, 3, Yes], [1, 22, 2, 2, Yes], [1, 23, 1, 1, Yes], [1, 24, 0, 0, Yes], [2, 0, 23, 23, Yes], [2, 1, 22, 22, Yes], [2, 2, 21, 21, Yes], [2, 3, 20, 20, Yes], [2, 4, 19, 19, Yes], [2, 5, 18, 18, Yes], [2, 6, 17, 17, Yes], [2, 7, 16, 16, Yes], [2, 8, 15, 15, Yes], [2, 9, 14, 14, Yes], [2, 10, 13, 13, Yes], [2, 11, 12, 12, Yes], [2, 12, 11, 11, Yes], [2, 13, 10, 10, Yes], [2, 14, 9, 9, Yes], [2, 15, 8, 8, Yes], [2, 16, 7, 7, Yes], [2, 17, 6, 6, Yes], [2, 18, 5, 5, Yes], [2, 19, 4, 4, Yes], [2, 20, 3, 3, Yes], [2, 21, 2, 2, Yes], [2, 22, 1, 1, Yes], [2, 23, 0, 0, Yes], [2, 24, 24, 24, Yes], [3, 0, 22, 22, Yes], [3, 1, 21, 21, Yes], [3, 2, 20, 20, Yes], [3, 3, 19, 19, Yes], [3, 4, 18, 18, Yes], [3, 5, 17, 17, Yes], [3, 6, 16, 16, Yes], [3, 7, 15, 15, Yes], [3, 8, 14, 14, Yes], [3, 9, 13, 13, Yes], [3, 10, 12, 12, Yes], [3, 11, 11, 11, Yes], [3, 12, 10, 10, Yes], [3, 13, 9, 9, Yes], [3, 14, 8, 8, Yes], [3, 15, 7, 7, Yes], [3, 16, 6, 6, Yes], [3, 17, 5, 5, Yes], [3, 18, 4, 4, Yes], [3, 19, 3, 3, Yes], [3, 20, 2, 2, Yes], [3, 21, 1, 1, Yes], [3, 22, 0, 0, Yes], [3, 23, 24, 24, Yes], [3, 24, 23, 23, Yes], [4, 0, 21, 21, Yes], [4, 1, 20, 20, Yes], [4, 2, 19, 19, Yes], [4, 3, 18, 18, Yes], [4, 4, 17, 17, Yes], [4, 5, 16, 16, Yes], [4, 6, 15, 15, Yes], [4, 7, 14, 14, Yes], [4, 8, 13, 13, Yes], [4, 9, 12, 12, Yes], [4, 10, 11, 11, Yes], [4, 11, 10, 10, Yes], [4, 12, 9, 9, Yes], [4, 13, 8, 8, Yes], [4, 14, 7, 7, Yes], [4, 15, 6, 6, Yes], [4, 16, 5, 5, Yes], [4, 17, 4, 4, Yes], [4, 18, 3, 3, Yes], [4, 19, 2, 2, Yes], [4, 20, 1, 1, Yes], [4, 21, 0, 0, Yes], [4, 22, 24, 24, Yes], [4, 23, 23, 23, Yes], [4, 24, 22, 22, Yes], [5, 0, 20, 20, Yes], [5, 1, 19, 19, Yes], [5, 2, 18, 18, Yes], [5, 3, 17, 17, Yes], [5, 4, 16, 16, Yes], [5, 5, 15, 15, Yes], [5, 6, 14, 14, Yes], [5, 7, 13, 13, Yes], [5, 8, 12, 12, Yes], [5, 9, 11, 11, Yes], [5, 10, 10, 10, Yes], [5, 11, 9, 9, Yes], [5, 12, 8, 8, Yes], [5, 13, 7, 7, Yes], [5, 14, 6, 6, Yes], [5, 15, 5, 5, Yes], [5, 16, 4, 4, Yes], [5, 17, 3, 3, Yes], [5, 18, 2, 2, Yes], [5, 19, 1, 1, Yes], [5, 20, 0, 0, Yes], [5, 21, 24, 24, Yes], [5, 22, 23, 23, Yes], [5, 23, 22, 22, Yes], [5, 24, 21, 21, Yes], [6, 0, 19, 19, Yes], [6, 1, 18, 18, Yes], [6, 2, 17, 17, Yes], [6, 3, 16, 16, Yes], [6, 4, 15, 15, Yes], [6, 5, 14, 14, Yes], [6, 6, 13, 13, Yes], [6, 7, 12, 12, Yes], [6, 8, 11, 11, Yes], [6, 9, 10, 10, Yes], [6, 10, 9, 9, Yes], [6, 11, 8, 8, Yes], [6, 12, 7, 7, Yes], [6, 13, 6, 6, Yes], [6, 14, 5, 5, Yes], [6, 15, 4, 4, Yes], [6, 16, 3, 3, Yes], [6, 17, 2, 2, Yes], [6, 18, 1, 1, Yes], [6, 19, 0, 0, Yes], [6, 20, 24, 24, Yes], [6, 21, 23, 23, Yes], [6, 22, 22, 22, Yes], [6, 23, 21, 21, Yes], [6, 24, 20, 20, Yes], [7, 0, 18, 18, Yes], [7, 1, 17, 17, Yes], [7, 2, 16, 16, Yes], [7, 3, 15, 15, Yes], [7, 4, 14, 14, Yes], [7, 5, 13, 13, Yes], [7, 6, 12, 12, Yes], [7, 7, 11, 11, Yes], [7, 8, 10, 10, Yes], [7, 9, 9, 9, Yes], [7, 10, 8, 8, Yes], [7, 11, 7, 7, Yes], [7, 12, 6, 6, Yes], [7, 13, 5, 5, Yes], [7, 14, 4, 4, Yes], [7, 15, 3, 3, Yes], [7, 16, 2, 2, Yes], [7, 17, 1, 1, Yes], [7, 18, 0, 0, Yes], [7, 19, 24, 24, Yes], [7, 20, 23, 23, Yes], [7, 21, 22, 22, Yes], [7, 22, 21, 21, Yes], [7, 23, 20, 20, Yes], [7, 24, 19, 19, Yes], [8, 0, 17, 17, Yes], [8, 1, 16, 16, Yes], [8, 2, 15, 15, Yes], [8, 3, 14, 14, Yes], [8, 4, 13, 13, Yes], [8, 5, 12, 12, Yes], [8, 6, 11, 11, Yes], [8, 7, 10, 10, Yes], [8, 8, 9, 9, Yes], [8, 9, 8, 8, Yes], [8, 10, 7, 7, Yes], [8, 11, 6, 6, Yes], [8, 12, 5, 5, Yes], [8, 13, 4, 4, Yes], [8, 14, 3, 3, Yes], [8, 15, 2, 2, Yes], [8, 16, 1, 1, Yes], [8, 17, 0, 0, Yes], [8, 18, 24, 24, Yes], [8, 19, 23, 23, Yes], [8, 20, 22, 22, Yes], [8, 21, 21, 21, Yes], [8, 22, 20, 20, Yes], [8, 23, 19, 19, Yes], [8, 24, 18, 18, Yes], [9, 0, 16, 16, Yes], [9, 1, 15, 15, Yes], [9, 2, 14, 14, Yes], [9, 3, 13, 13, Yes], [9, 4, 12, 12, Yes], [9, 5, 11, 11, Yes], [9, 6, 10, 10, Yes], [9, 7, 9, 9, Yes], [9, 8, 8, 8, Yes], [9, 9, 7, 7, Yes], [9, 10, 6, 6, Yes], [9, 11, 5, 5, Yes], [9, 12, 4, 4, Yes], [9, 13, 3, 3, Yes], [9, 14, 2, 2, Yes], [9, 15, 1, 1, Yes], [9, 16, 0, 0, Yes], [9, 17, 24, 24, Yes], [9, 18, 23, 23, Yes], [9, 19, 22, 22, Yes], [9, 20, 21, 21, Yes], [9, 21, 20, 20, Yes], [9, 22, 19, 19, Yes], [9, 23, 18, 18, Yes], [9, 24, 17, 17, Yes], [10, 0, 15, 15, Yes], [10, 1, 14, 14, Yes], [10, 2, 13, 13, Yes], [10, 3, 12, 12, Yes], [10, 4, 11, 11, Yes], [10, 5, 10, 10, Yes], [10, 6, 9, 9, Yes], [10, 7, 8, 8, Yes], [10, 8, 7, 7, Yes], [10, 9, 6, 6, Yes], [10, 10, 5, 5, Yes], [10, 11, 4, 4, Yes], [10, 12, 3, 3, Yes], [10, 13, 2, 2, Yes], [10, 14, 1, 1, Yes], [10, 15, 0, 0, Yes], [10, 16, 24, 24, Yes], [10, 17, 23, 23, Yes], [10, 18, 22, 22, Yes], [10, 19, 21, 21, Yes], [10, 20, 20, 20, Yes], [10, 21, 19, 19, Yes], [10, 22, 18, 18, Yes], [10, 23, 17, 17, Yes], [10, 24, 16, 16, Yes], [11, 0, 14, 14, Yes], [11, 1, 13, 13, Yes], [11, 2, 12, 12, Yes], [11, 3, 11, 11, Yes], [11, 4, 10, 10, Yes], [11, 5, 9, 9, Yes], [11, 6, 8, 8, Yes], [11, 7, 7, 7, Yes], [11, 8, 6, 6, Yes], [11, 9, 5, 5, Yes], [11, 10, 4, 4, Yes], [11, 11, 3, 3, Yes], [11, 12, 2, 2, Yes], [11, 13, 1, 1, Yes], [11, 14, 0, 0, Yes], [11, 15, 24, 24, Yes], [11, 16, 23, 23, Yes], [11, 17, 22, 22, Yes], [11, 18, 21, 21, Yes], [11, 19, 20, 20, Yes], [11, 20, 19, 19, Yes], [11, 21, 18, 18, Yes], [11, 22, 17, 17, Yes], [11, 23, 16, 16, Yes], [11, 24, 15, 15, Yes], [12, 0, 13, 13, Yes], [12, 1, 12, 12, Yes], [12, 2, 11, 11, Yes], [12, 3, 10, 10, Yes], [12, 4, 9, 9, Yes], [12, 5, 8, 8, Yes], [12, 6, 7, 7, Yes], [12, 7, 6, 6, Yes], [12, 8, 5, 5, Yes], [12, 9, 4, 4, Yes], [12, 10, 3, 3, Yes], [12, 11, 2, 2, Yes], [12, 12, 1, 1, Yes], [12, 13, 0, 0, Yes], [12, 14, 24, 24, Yes], [12, 15, 23, 23, Yes], [12, 16, 22, 22, Yes], [12, 17, 21, 21, Yes], [12, 18, 20, 20, Yes], [12, 19, 19, 19, Yes], [12, 20, 18, 18, Yes], [12, 21, 17, 17, Yes], [12, 22, 16, 16, Yes], [12, 23, 15, 15, Yes], [12, 24, 14, 14, Yes], [13, 0, 12, 12, Yes], [13, 1, 11, 11, Yes], [13, 2, 10, 10, Yes], [13, 3, 9, 9, Yes], [13, 4, 8, 8, Yes], [13, 5, 7, 7, Yes], [13, 6, 6, 6, Yes], [13, 7, 5, 5, Yes], [13, 8, 4, 4, Yes], [13, 9, 3, 3, Yes], [13, 10, 2, 2, Yes], [13, 11, 1, 1, Yes], [13, 12, 0, 0, Yes], [13, 13, 24, 24, Yes], [13, 14, 23, 23, Yes], [13, 15, 22, 22, Yes], [13, 16, 21, 21, Yes], [13, 17, 20, 20, Yes], [13, 18, 19, 19, Yes], [13, 19, 18, 18, Yes], [13, 20, 17, 17, Yes], [13, 21, 16, 16, Yes], [13, 22, 15, 15, Yes], [13, 23, 14, 14, Yes], [13, 24, 13, 13, Yes], [14, 0, 11, 11, Yes], [14, 1, 10, 10, Yes], [14, 2, 9, 9, Yes], [14, 3, 8, 8, Yes], [14, 4, 7, 7, Yes], [14, 5, 6, 6, Yes], [14, 6, 5, 5, Yes], [14, 7, 4, 4, Yes], [14, 8, 3, 3, Yes], [14, 9, 2, 2, Yes], [14, 10, 1, 1, Yes], [14, 11, 0, 0, Yes], [14, 12, 24, 24, Yes], [14, 13, 23, 23, Yes], [14, 14, 22, 22, Yes], [14, 15, 21, 21, Yes], [14, 16, 20, 20, Yes], [14, 17, 19, 19, Yes], [14, 18, 18, 18, Yes], [14, 19, 17, 17, Yes], [14, 20, 16, 16, Yes], [14, 21, 15, 15, Yes], [14, 22, 14, 14, Yes], [14, 23, 13, 13, Yes], [14, 24, 12, 12, Yes], [15, 0, 10, 10, Yes], [15, 1, 9, 9, Yes], [15, 2, 8, 8, Yes], [15, 3, 7, 7, Yes], [15, 4, 6, 6, Yes], [15, 5, 5, 5, Yes], [15, 6, 4, 4, Yes], [15, 7, 3, 3, Yes], [15, 8, 2, 2, Yes], [15, 9, 1, 1, Yes], [15, 10, 0, 0, Yes], [15, 11, 24, 24, Yes], [15, 12, 23, 23, Yes], [15, 13, 22, 22, Yes], [15, 14, 21, 21, Yes], [15, 15, 20, 20, Yes], [15, 16, 19, 19, Yes], [15, 17, 18, 18, Yes], [15, 18, 17, 17, Yes], [15, 19, 16, 16, Yes], [15, 20, 15, 15, Yes], [15, 21, 14, 14, Yes], [15, 22, 13, 13, Yes], [15, 23, 12, 12, Yes], [15, 24, 11, 11, Yes], [16, 0, 9, 9, Yes], [16, 1, 8, 8, Yes], [16, 2, 7, 7, Yes], [16, 3, 6, 6, Yes], [16, 4, 5, 5, Yes], [16, 5, 4, 4, Yes], [16, 6, 3, 3, Yes], [16, 7, 2, 2, Yes], [16, 8, 1, 1, Yes], [16, 9, 0, 0, Yes], [16, 10, 24, 24, Yes], [16, 11, 23, 23, Yes], [16, 12, 22, 22, Yes], [16, 13, 21, 21, Yes], [16, 14, 20, 20, Yes], [16, 15, 19, 19, Yes], [16, 16, 18, 18, Yes], [16, 17, 17, 17, Yes], [16, 18, 16, 16, Yes], [16, 19, 15, 15, Yes], [16, 20, 14, 14, Yes], [16, 21, 13, 13, Yes], [16, 22, 12, 12, Yes], [16, 23, 11, 11, Yes], [16, 24, 10, 10, Yes], [17, 0, 8, 8, Yes], [17, 1, 7, 7, Yes], [17, 2, 6, 6, Yes], [17, 3, 5, 5, Yes], [17, 4, 4, 4, Yes], [17, 5, 3, 3, Yes], [17, 6, 2, 2, Yes], [17, 7, 1, 1, Yes], [17, 8, 0, 0, Yes], [17, 9, 24, 24, Yes], [17, 10, 23, 23, Yes], [17, 11, 22, 22, Yes], [17, 12, 21, 21, Yes], [17, 13, 20, 20, Yes], [17, 14, 19, 19, Yes], [17, 15, 18, 18, Yes], [17, 16, 17, 17, Yes], [17, 17, 16, 16, Yes], [17, 18, 15, 15, Yes], [17, 19, 14, 14, Yes], [17, 20, 13, 13, Yes], [17, 21, 12, 12, Yes], [17, 22, 11, 11, Yes], [17, 23, 10, 10, Yes], [17, 24, 9, 9, Yes], [18, 0, 7, 7, Yes], [18, 1, 6, 6, Yes], [18, 2, 5, 5, Yes], [18, 3, 4, 4, Yes], [18, 4, 3, 3, Yes], [18, 5, 2, 2, Yes], [18, 6, 1, 1, Yes], [18, 7, 0, 0, Yes], [18, 8, 24, 24, Yes], [18, 9, 23, 23, Yes], [18, 10, 22, 22, Yes], [18, 11, 21, 21, Yes], [18, 12, 20, 20, Yes], [18, 13, 19, 19, Yes], [18, 14, 18, 18, Yes], [18, 15, 17, 17, Yes], [18, 16, 16, 16, Yes], [18, 17, 15, 15, Yes], [18, 18, 14, 14, Yes], [18, 19, 13, 13, Yes], [18, 20, 12, 12, Yes], [18, 21, 11, 11, Yes], [18, 22, 10, 10, Yes], [18, 23, 9, 9, Yes], [18, 24, 8, 8, Yes], [19, 0, 6, 6, Yes], [19, 1, 5, 5, Yes], [19, 2, 4, 4, Yes], [19, 3, 3, 3, Yes], [19, 4, 2, 2, Yes], [19, 5, 1, 1, Yes], [19, 6, 0, 0, Yes], [19, 7, 24, 24, Yes], [19, 8, 23, 23, Yes], [19, 9, 22, 22, Yes], [19, 10, 21, 21, Yes], [19, 11, 20, 20, Yes], [19, 12, 19, 19, Yes], [19, 13, 18, 18, Yes], [19, 14, 17, 17, Yes], [19, 15, 16, 16, Yes], [19, 16, 15, 15, Yes], [19, 17, 14, 14, Yes], [19, 18, 13, 13, Yes], [19, 19, 12, 12, Yes], [19, 20, 11, 11, Yes], [19, 21, 10, 10, Yes], [19, 22, 9, 9, Yes], [19, 23, 8, 8, Yes], [19, 24, 7, 7, Yes], [20, 0, 5, 5, Yes], [20, 1, 4, 4, Yes], [20, 2, 3, 3, Yes], [20, 3, 2, 2, Yes], [20, 4, 1, 1, Yes], [20, 5, 0, 0, Yes], [20, 6, 24, 24, Yes], [20, 7, 23, 23, Yes], [20, 8, 22, 22, Yes], [20, 9, 21, 21, Yes], [20, 10, 20, 20, Yes], [20, 11, 19, 19, Yes], [20, 12, 18, 18, Yes], [20, 13, 17, 17, Yes], [20, 14, 16, 16, Yes], [20, 15, 15, 15, Yes], [20, 16, 14, 14, Yes], [20, 17, 13, 13, Yes], [20, 18, 12, 12, Yes], [20, 19, 11, 11, Yes], [20, 20, 10, 10, Yes], [20, 21, 9, 9, Yes], [20, 22, 8, 8, Yes], [20, 23, 7, 7, Yes], [20, 24, 6, 6, Yes], [21, 0, 4, 4, Yes], [21, 1, 3, 3, Yes], [21, 2, 2, 2, Yes], [21, 3, 1, 1, Yes], [21, 4, 0, 0, Yes], [21, 5, 24, 24, Yes], [21, 6, 23, 23, Yes], [21, 7, 22, 22, Yes], [21, 8, 21, 21, Yes], [21, 9, 20, 20, Yes], [21, 10, 19, 19, Yes], [21, 11, 18, 18, Yes], [21, 12, 17, 17, Yes], [21, 13, 16, 16, Yes], [21, 14, 15, 15, Yes], [21, 15, 14, 14, Yes], [21, 16, 13, 13, Yes], [21, 17, 12, 12, Yes], [21, 18, 11, 11, Yes], [21, 19, 10, 10, Yes], [21, 20, 9, 9, Yes], [21, 21, 8, 8, Yes], [21, 22, 7, 7, Yes], [21, 23, 6, 6, Yes], [21, 24, 5, 5, Yes], [22, 0, 3, 3, Yes], [22, 1, 2, 2, Yes], [22, 2, 1, 1, Yes], [22, 3, 0, 0, Yes], [22, 4, 24, 24, Yes], [22, 5, 23, 23, Yes], [22, 6, 22, 22, Yes], [22, 7, 21, 21, Yes], [22, 8, 20, 20, Yes], [22, 9, 19, 19, Yes], [22, 10, 18, 18, Yes], [22, 11, 17, 17, Yes], [22, 12, 16, 16, Yes], [22, 13, 15, 15, Yes], [22, 14, 14, 14, Yes], [22, 15, 13, 13, Yes], [22, 16, 12, 12, Yes], [22, 17, 11, 11, Yes], [22, 18, 10, 10, Yes], [22, 19, 9, 9, Yes], [22, 20, 8, 8, Yes], [22, 21, 7, 7, Yes], [22, 22, 6, 6, Yes], [22, 23, 5, 5, Yes], [22, 24, 4, 4, Yes], [23, 0, 2, 2, Yes], [23, 1, 1, 1, Yes], [23, 2, 0, 0, Yes], [23, 3, 24, 24, Yes], [23, 4, 23, 23, Yes], [23, 5, 22, 22, Yes], [23, 6, 21, 21, Yes], [23, 7, 20, 20, Yes], [23, 8, 19, 19, Yes], [23, 9, 18, 18, Yes], [23, 10, 17, 17, Yes], [23, 11, 16, 16, Yes], [23, 12, 15, 15, Yes], [23, 13, 14, 14, Yes], [23, 14, 13, 13, Yes], [23, 15, 12, 12, Yes], [23, 16, 11, 11, Yes], [23, 17, 10, 10, Yes], [23, 18, 9, 9, Yes], [23, 19, 8, 8, Yes], [23, 20, 7, 7, Yes], [23, 21, 6, 6, Yes], [23, 22, 5, 5, Yes], [23, 23, 4, 4, Yes], [23, 24, 3, 3, Yes], [24, 0, 1, 1, Yes], [24, 1, 0, 0, Yes], [24, 2, 24, 24, Yes], [24, 3, 23, 23, Yes], [24, 4, 22, 22, Yes], [24, 5, 21, 21, Yes], [24, 6, 20, 20, Yes], [24, 7, 19, 19, Yes], [24, 8, 18, 18, Yes], [24, 9, 17, 17, Yes], [24, 10, 16, 16, Yes], [24, 11, 15, 15, Yes], [24, 12, 14, 14, Yes], [24, 13, 13, 13, Yes], [24, 14, 12, 12, Yes], [24, 15, 11, 11, Yes], [24, 16, 10, 10, Yes], [24, 17, 9, 9, Yes], [24, 18, 8, 8, Yes], [24, 19, 7, 7, Yes], [24, 20, 6, 6, Yes], [24, 21, 5, 5, Yes], [24, 22, 4, 4, Yes], [24, 23, 3, 3, Yes], [24, 24, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0024","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9293c1b17fc84c2442f4cccb9618d52ef047a7d4179039ca844db1b0be6e2632","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141024,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], target identity=0, and total mapping=[0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(26, 26, [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 25, 25, Yes], [0, 2, 24, 24, Yes], [0, 3, 23, 23, Yes], [0, 4, 22, 22, Yes], [0, 5, 21, 21, Yes], [0, 6, 20, 20, Yes], [0, 7, 19, 19, Yes], [0, 8, 18, 18, Yes], [0, 9, 17, 17, Yes], [0, 10, 16, 16, Yes], [0, 11, 15, 15, Yes], [0, 12, 14, 14, Yes], [0, 13, 13, 13, Yes], [0, 14, 12, 12, Yes], [0, 15, 11, 11, Yes], [0, 16, 10, 10, Yes], [0, 17, 9, 9, Yes], [0, 18, 8, 8, Yes], [0, 19, 7, 7, Yes], [0, 20, 6, 6, Yes], [0, 21, 5, 5, Yes], [0, 22, 4, 4, Yes], [0, 23, 3, 3, Yes], [0, 24, 2, 2, Yes], [0, 25, 1, 1, Yes], [1, 0, 25, 25, Yes], [1, 1, 24, 24, Yes], [1, 2, 23, 23, Yes], [1, 3, 22, 22, Yes], [1, 4, 21, 21, Yes], [1, 5, 20, 20, Yes], [1, 6, 19, 19, Yes], [1, 7, 18, 18, Yes], [1, 8, 17, 17, Yes], [1, 9, 16, 16, Yes], [1, 10, 15, 15, Yes], [1, 11, 14, 14, Yes], [1, 12, 13, 13, Yes], [1, 13, 12, 12, Yes], [1, 14, 11, 11, Yes], [1, 15, 10, 10, Yes], [1, 16, 9, 9, Yes], [1, 17, 8, 8, Yes], [1, 18, 7, 7, Yes], [1, 19, 6, 6, Yes], [1, 20, 5, 5, Yes], [1, 21, 4, 4, Yes], [1, 22, 3, 3, Yes], [1, 23, 2, 2, Yes], [1, 24, 1, 1, Yes], [1, 25, 0, 0, Yes], [2, 0, 24, 24, Yes], [2, 1, 23, 23, Yes], [2, 2, 22, 22, Yes], [2, 3, 21, 21, Yes], [2, 4, 20, 20, Yes], [2, 5, 19, 19, Yes], [2, 6, 18, 18, Yes], [2, 7, 17, 17, Yes], [2, 8, 16, 16, Yes], [2, 9, 15, 15, Yes], [2, 10, 14, 14, Yes], [2, 11, 13, 13, Yes], [2, 12, 12, 12, Yes], [2, 13, 11, 11, Yes], [2, 14, 10, 10, Yes], [2, 15, 9, 9, Yes], [2, 16, 8, 8, Yes], [2, 17, 7, 7, Yes], [2, 18, 6, 6, Yes], [2, 19, 5, 5, Yes], [2, 20, 4, 4, Yes], [2, 21, 3, 3, Yes], [2, 22, 2, 2, Yes], [2, 23, 1, 1, Yes], [2, 24, 0, 0, Yes], [2, 25, 25, 25, Yes], [3, 0, 23, 23, Yes], [3, 1, 22, 22, Yes], [3, 2, 21, 21, Yes], [3, 3, 20, 20, Yes], [3, 4, 19, 19, Yes], [3, 5, 18, 18, Yes], [3, 6, 17, 17, Yes], [3, 7, 16, 16, Yes], [3, 8, 15, 15, Yes], [3, 9, 14, 14, Yes], [3, 10, 13, 13, Yes], [3, 11, 12, 12, Yes], [3, 12, 11, 11, Yes], [3, 13, 10, 10, Yes], [3, 14, 9, 9, Yes], [3, 15, 8, 8, Yes], [3, 16, 7, 7, Yes], [3, 17, 6, 6, Yes], [3, 18, 5, 5, Yes], [3, 19, 4, 4, Yes], [3, 20, 3, 3, Yes], [3, 21, 2, 2, Yes], [3, 22, 1, 1, Yes], [3, 23, 0, 0, Yes], [3, 24, 25, 25, Yes], [3, 25, 24, 24, Yes], [4, 0, 22, 22, Yes], [4, 1, 21, 21, Yes], [4, 2, 20, 20, Yes], [4, 3, 19, 19, Yes], [4, 4, 18, 18, Yes], [4, 5, 17, 17, Yes], [4, 6, 16, 16, Yes], [4, 7, 15, 15, Yes], [4, 8, 14, 14, Yes], [4, 9, 13, 13, Yes], [4, 10, 12, 12, Yes], [4, 11, 11, 11, Yes], [4, 12, 10, 10, Yes], [4, 13, 9, 9, Yes], [4, 14, 8, 8, Yes], [4, 15, 7, 7, Yes], [4, 16, 6, 6, Yes], [4, 17, 5, 5, Yes], [4, 18, 4, 4, Yes], [4, 19, 3, 3, Yes], [4, 20, 2, 2, Yes], [4, 21, 1, 1, Yes], [4, 22, 0, 0, Yes], [4, 23, 25, 25, Yes], [4, 24, 24, 24, Yes], [4, 25, 23, 23, Yes], [5, 0, 21, 21, Yes], [5, 1, 20, 20, Yes], [5, 2, 19, 19, Yes], [5, 3, 18, 18, Yes], [5, 4, 17, 17, Yes], [5, 5, 16, 16, Yes], [5, 6, 15, 15, Yes], [5, 7, 14, 14, Yes], [5, 8, 13, 13, Yes], [5, 9, 12, 12, Yes], [5, 10, 11, 11, Yes], [5, 11, 10, 10, Yes], [5, 12, 9, 9, Yes], [5, 13, 8, 8, Yes], [5, 14, 7, 7, Yes], [5, 15, 6, 6, Yes], [5, 16, 5, 5, Yes], [5, 17, 4, 4, Yes], [5, 18, 3, 3, Yes], [5, 19, 2, 2, Yes], [5, 20, 1, 1, Yes], [5, 21, 0, 0, Yes], [5, 22, 25, 25, Yes], [5, 23, 24, 24, Yes], [5, 24, 23, 23, Yes], [5, 25, 22, 22, Yes], [6, 0, 20, 20, Yes], [6, 1, 19, 19, Yes], [6, 2, 18, 18, Yes], [6, 3, 17, 17, Yes], [6, 4, 16, 16, Yes], [6, 5, 15, 15, Yes], [6, 6, 14, 14, Yes], [6, 7, 13, 13, Yes], [6, 8, 12, 12, Yes], [6, 9, 11, 11, Yes], [6, 10, 10, 10, Yes], [6, 11, 9, 9, Yes], [6, 12, 8, 8, Yes], [6, 13, 7, 7, Yes], [6, 14, 6, 6, Yes], [6, 15, 5, 5, Yes], [6, 16, 4, 4, Yes], [6, 17, 3, 3, Yes], [6, 18, 2, 2, Yes], [6, 19, 1, 1, Yes], [6, 20, 0, 0, Yes], [6, 21, 25, 25, Yes], [6, 22, 24, 24, Yes], [6, 23, 23, 23, Yes], [6, 24, 22, 22, Yes], [6, 25, 21, 21, Yes], [7, 0, 19, 19, Yes], [7, 1, 18, 18, Yes], [7, 2, 17, 17, Yes], [7, 3, 16, 16, Yes], [7, 4, 15, 15, Yes], [7, 5, 14, 14, Yes], [7, 6, 13, 13, Yes], [7, 7, 12, 12, Yes], [7, 8, 11, 11, Yes], [7, 9, 10, 10, Yes], [7, 10, 9, 9, Yes], [7, 11, 8, 8, Yes], [7, 12, 7, 7, Yes], [7, 13, 6, 6, Yes], [7, 14, 5, 5, Yes], [7, 15, 4, 4, Yes], [7, 16, 3, 3, Yes], [7, 17, 2, 2, Yes], [7, 18, 1, 1, Yes], [7, 19, 0, 0, Yes], [7, 20, 25, 25, Yes], [7, 21, 24, 24, Yes], [7, 22, 23, 23, Yes], [7, 23, 22, 22, Yes], [7, 24, 21, 21, Yes], [7, 25, 20, 20, Yes], [8, 0, 18, 18, Yes], [8, 1, 17, 17, Yes], [8, 2, 16, 16, Yes], [8, 3, 15, 15, Yes], [8, 4, 14, 14, Yes], [8, 5, 13, 13, Yes], [8, 6, 12, 12, Yes], [8, 7, 11, 11, Yes], [8, 8, 10, 10, Yes], [8, 9, 9, 9, Yes], [8, 10, 8, 8, Yes], [8, 11, 7, 7, Yes], [8, 12, 6, 6, Yes], [8, 13, 5, 5, Yes], [8, 14, 4, 4, Yes], [8, 15, 3, 3, Yes], [8, 16, 2, 2, Yes], [8, 17, 1, 1, Yes], [8, 18, 0, 0, Yes], [8, 19, 25, 25, Yes], [8, 20, 24, 24, Yes], [8, 21, 23, 23, Yes], [8, 22, 22, 22, Yes], [8, 23, 21, 21, Yes], [8, 24, 20, 20, Yes], [8, 25, 19, 19, Yes], [9, 0, 17, 17, Yes], [9, 1, 16, 16, Yes], [9, 2, 15, 15, Yes], [9, 3, 14, 14, Yes], [9, 4, 13, 13, Yes], [9, 5, 12, 12, Yes], [9, 6, 11, 11, Yes], [9, 7, 10, 10, Yes], [9, 8, 9, 9, Yes], [9, 9, 8, 8, Yes], [9, 10, 7, 7, Yes], [9, 11, 6, 6, Yes], [9, 12, 5, 5, Yes], [9, 13, 4, 4, Yes], [9, 14, 3, 3, Yes], [9, 15, 2, 2, Yes], [9, 16, 1, 1, Yes], [9, 17, 0, 0, Yes], [9, 18, 25, 25, Yes], [9, 19, 24, 24, Yes], [9, 20, 23, 23, Yes], [9, 21, 22, 22, Yes], [9, 22, 21, 21, Yes], [9, 23, 20, 20, Yes], [9, 24, 19, 19, Yes], [9, 25, 18, 18, Yes], [10, 0, 16, 16, Yes], [10, 1, 15, 15, Yes], [10, 2, 14, 14, Yes], [10, 3, 13, 13, Yes], [10, 4, 12, 12, Yes], [10, 5, 11, 11, Yes], [10, 6, 10, 10, Yes], [10, 7, 9, 9, Yes], [10, 8, 8, 8, Yes], [10, 9, 7, 7, Yes], [10, 10, 6, 6, Yes], [10, 11, 5, 5, Yes], [10, 12, 4, 4, Yes], [10, 13, 3, 3, Yes], [10, 14, 2, 2, Yes], [10, 15, 1, 1, Yes], [10, 16, 0, 0, Yes], [10, 17, 25, 25, Yes], [10, 18, 24, 24, Yes], [10, 19, 23, 23, Yes], [10, 20, 22, 22, Yes], [10, 21, 21, 21, Yes], [10, 22, 20, 20, Yes], [10, 23, 19, 19, Yes], [10, 24, 18, 18, Yes], [10, 25, 17, 17, Yes], [11, 0, 15, 15, Yes], [11, 1, 14, 14, Yes], [11, 2, 13, 13, Yes], [11, 3, 12, 12, Yes], [11, 4, 11, 11, Yes], [11, 5, 10, 10, Yes], [11, 6, 9, 9, Yes], [11, 7, 8, 8, Yes], [11, 8, 7, 7, Yes], [11, 9, 6, 6, Yes], [11, 10, 5, 5, Yes], [11, 11, 4, 4, Yes], [11, 12, 3, 3, Yes], [11, 13, 2, 2, Yes], [11, 14, 1, 1, Yes], [11, 15, 0, 0, Yes], [11, 16, 25, 25, Yes], [11, 17, 24, 24, Yes], [11, 18, 23, 23, Yes], [11, 19, 22, 22, Yes], [11, 20, 21, 21, Yes], [11, 21, 20, 20, Yes], [11, 22, 19, 19, Yes], [11, 23, 18, 18, Yes], [11, 24, 17, 17, Yes], [11, 25, 16, 16, Yes], [12, 0, 14, 14, Yes], [12, 1, 13, 13, Yes], [12, 2, 12, 12, Yes], [12, 3, 11, 11, Yes], [12, 4, 10, 10, Yes], [12, 5, 9, 9, Yes], [12, 6, 8, 8, Yes], [12, 7, 7, 7, Yes], [12, 8, 6, 6, Yes], [12, 9, 5, 5, Yes], [12, 10, 4, 4, Yes], [12, 11, 3, 3, Yes], [12, 12, 2, 2, Yes], [12, 13, 1, 1, Yes], [12, 14, 0, 0, Yes], [12, 15, 25, 25, Yes], [12, 16, 24, 24, Yes], [12, 17, 23, 23, Yes], [12, 18, 22, 22, Yes], [12, 19, 21, 21, Yes], [12, 20, 20, 20, Yes], [12, 21, 19, 19, Yes], [12, 22, 18, 18, Yes], [12, 23, 17, 17, Yes], [12, 24, 16, 16, Yes], [12, 25, 15, 15, Yes], [13, 0, 13, 13, Yes], [13, 1, 12, 12, Yes], [13, 2, 11, 11, Yes], [13, 3, 10, 10, Yes], [13, 4, 9, 9, Yes], [13, 5, 8, 8, Yes], [13, 6, 7, 7, Yes], [13, 7, 6, 6, Yes], [13, 8, 5, 5, Yes], [13, 9, 4, 4, Yes], [13, 10, 3, 3, Yes], [13, 11, 2, 2, Yes], [13, 12, 1, 1, Yes], [13, 13, 0, 0, Yes], [13, 14, 25, 25, Yes], [13, 15, 24, 24, Yes], [13, 16, 23, 23, Yes], [13, 17, 22, 22, Yes], [13, 18, 21, 21, Yes], [13, 19, 20, 20, Yes], [13, 20, 19, 19, Yes], [13, 21, 18, 18, Yes], [13, 22, 17, 17, Yes], [13, 23, 16, 16, Yes], [13, 24, 15, 15, Yes], [13, 25, 14, 14, Yes], [14, 0, 12, 12, Yes], [14, 1, 11, 11, Yes], [14, 2, 10, 10, Yes], [14, 3, 9, 9, Yes], [14, 4, 8, 8, Yes], [14, 5, 7, 7, Yes], [14, 6, 6, 6, Yes], [14, 7, 5, 5, Yes], [14, 8, 4, 4, Yes], [14, 9, 3, 3, Yes], [14, 10, 2, 2, Yes], [14, 11, 1, 1, Yes], [14, 12, 0, 0, Yes], [14, 13, 25, 25, Yes], [14, 14, 24, 24, Yes], [14, 15, 23, 23, Yes], [14, 16, 22, 22, Yes], [14, 17, 21, 21, Yes], [14, 18, 20, 20, Yes], [14, 19, 19, 19, Yes], [14, 20, 18, 18, Yes], [14, 21, 17, 17, Yes], [14, 22, 16, 16, Yes], [14, 23, 15, 15, Yes], [14, 24, 14, 14, Yes], [14, 25, 13, 13, Yes], [15, 0, 11, 11, Yes], [15, 1, 10, 10, Yes], [15, 2, 9, 9, Yes], [15, 3, 8, 8, Yes], [15, 4, 7, 7, Yes], [15, 5, 6, 6, Yes], [15, 6, 5, 5, Yes], [15, 7, 4, 4, Yes], [15, 8, 3, 3, Yes], [15, 9, 2, 2, Yes], [15, 10, 1, 1, Yes], [15, 11, 0, 0, Yes], [15, 12, 25, 25, Yes], [15, 13, 24, 24, Yes], [15, 14, 23, 23, Yes], [15, 15, 22, 22, Yes], [15, 16, 21, 21, Yes], [15, 17, 20, 20, Yes], [15, 18, 19, 19, Yes], [15, 19, 18, 18, Yes], [15, 20, 17, 17, Yes], [15, 21, 16, 16, Yes], [15, 22, 15, 15, Yes], [15, 23, 14, 14, Yes], [15, 24, 13, 13, Yes], [15, 25, 12, 12, Yes], [16, 0, 10, 10, Yes], [16, 1, 9, 9, Yes], [16, 2, 8, 8, Yes], [16, 3, 7, 7, Yes], [16, 4, 6, 6, Yes], [16, 5, 5, 5, Yes], [16, 6, 4, 4, Yes], [16, 7, 3, 3, Yes], [16, 8, 2, 2, Yes], [16, 9, 1, 1, Yes], [16, 10, 0, 0, Yes], [16, 11, 25, 25, Yes], [16, 12, 24, 24, Yes], [16, 13, 23, 23, Yes], [16, 14, 22, 22, Yes], [16, 15, 21, 21, Yes], [16, 16, 20, 20, Yes], [16, 17, 19, 19, Yes], [16, 18, 18, 18, Yes], [16, 19, 17, 17, Yes], [16, 20, 16, 16, Yes], [16, 21, 15, 15, Yes], [16, 22, 14, 14, Yes], [16, 23, 13, 13, Yes], [16, 24, 12, 12, Yes], [16, 25, 11, 11, Yes], [17, 0, 9, 9, Yes], [17, 1, 8, 8, Yes], [17, 2, 7, 7, Yes], [17, 3, 6, 6, Yes], [17, 4, 5, 5, Yes], [17, 5, 4, 4, Yes], [17, 6, 3, 3, Yes], [17, 7, 2, 2, Yes], [17, 8, 1, 1, Yes], [17, 9, 0, 0, Yes], [17, 10, 25, 25, Yes], [17, 11, 24, 24, Yes], [17, 12, 23, 23, Yes], [17, 13, 22, 22, Yes], [17, 14, 21, 21, Yes], [17, 15, 20, 20, Yes], [17, 16, 19, 19, Yes], [17, 17, 18, 18, Yes], [17, 18, 17, 17, Yes], [17, 19, 16, 16, Yes], [17, 20, 15, 15, Yes], [17, 21, 14, 14, Yes], [17, 22, 13, 13, Yes], [17, 23, 12, 12, Yes], [17, 24, 11, 11, Yes], [17, 25, 10, 10, Yes], [18, 0, 8, 8, Yes], [18, 1, 7, 7, Yes], [18, 2, 6, 6, Yes], [18, 3, 5, 5, Yes], [18, 4, 4, 4, Yes], [18, 5, 3, 3, Yes], [18, 6, 2, 2, Yes], [18, 7, 1, 1, Yes], [18, 8, 0, 0, Yes], [18, 9, 25, 25, Yes], [18, 10, 24, 24, Yes], [18, 11, 23, 23, Yes], [18, 12, 22, 22, Yes], [18, 13, 21, 21, Yes], [18, 14, 20, 20, Yes], [18, 15, 19, 19, Yes], [18, 16, 18, 18, Yes], [18, 17, 17, 17, Yes], [18, 18, 16, 16, Yes], [18, 19, 15, 15, Yes], [18, 20, 14, 14, Yes], [18, 21, 13, 13, Yes], [18, 22, 12, 12, Yes], [18, 23, 11, 11, Yes], [18, 24, 10, 10, Yes], [18, 25, 9, 9, Yes], [19, 0, 7, 7, Yes], [19, 1, 6, 6, Yes], [19, 2, 5, 5, Yes], [19, 3, 4, 4, Yes], [19, 4, 3, 3, Yes], [19, 5, 2, 2, Yes], [19, 6, 1, 1, Yes], [19, 7, 0, 0, Yes], [19, 8, 25, 25, Yes], [19, 9, 24, 24, Yes], [19, 10, 23, 23, Yes], [19, 11, 22, 22, Yes], [19, 12, 21, 21, Yes], [19, 13, 20, 20, Yes], [19, 14, 19, 19, Yes], [19, 15, 18, 18, Yes], [19, 16, 17, 17, Yes], [19, 17, 16, 16, Yes], [19, 18, 15, 15, Yes], [19, 19, 14, 14, Yes], [19, 20, 13, 13, Yes], [19, 21, 12, 12, Yes], [19, 22, 11, 11, Yes], [19, 23, 10, 10, Yes], [19, 24, 9, 9, Yes], [19, 25, 8, 8, Yes], [20, 0, 6, 6, Yes], [20, 1, 5, 5, Yes], [20, 2, 4, 4, Yes], [20, 3, 3, 3, Yes], [20, 4, 2, 2, Yes], [20, 5, 1, 1, Yes], [20, 6, 0, 0, Yes], [20, 7, 25, 25, Yes], [20, 8, 24, 24, Yes], [20, 9, 23, 23, Yes], [20, 10, 22, 22, Yes], [20, 11, 21, 21, Yes], [20, 12, 20, 20, Yes], [20, 13, 19, 19, Yes], [20, 14, 18, 18, Yes], [20, 15, 17, 17, Yes], [20, 16, 16, 16, Yes], [20, 17, 15, 15, Yes], [20, 18, 14, 14, Yes], [20, 19, 13, 13, Yes], [20, 20, 12, 12, Yes], [20, 21, 11, 11, Yes], [20, 22, 10, 10, Yes], [20, 23, 9, 9, Yes], [20, 24, 8, 8, Yes], [20, 25, 7, 7, Yes], [21, 0, 5, 5, Yes], [21, 1, 4, 4, Yes], [21, 2, 3, 3, Yes], [21, 3, 2, 2, Yes], [21, 4, 1, 1, Yes], [21, 5, 0, 0, Yes], [21, 6, 25, 25, Yes], [21, 7, 24, 24, Yes], [21, 8, 23, 23, Yes], [21, 9, 22, 22, Yes], [21, 10, 21, 21, Yes], [21, 11, 20, 20, Yes], [21, 12, 19, 19, Yes], [21, 13, 18, 18, Yes], [21, 14, 17, 17, Yes], [21, 15, 16, 16, Yes], [21, 16, 15, 15, Yes], [21, 17, 14, 14, Yes], [21, 18, 13, 13, Yes], [21, 19, 12, 12, Yes], [21, 20, 11, 11, Yes], [21, 21, 10, 10, Yes], [21, 22, 9, 9, Yes], [21, 23, 8, 8, Yes], [21, 24, 7, 7, Yes], [21, 25, 6, 6, Yes], [22, 0, 4, 4, Yes], [22, 1, 3, 3, Yes], [22, 2, 2, 2, Yes], [22, 3, 1, 1, Yes], [22, 4, 0, 0, Yes], [22, 5, 25, 25, Yes], [22, 6, 24, 24, Yes], [22, 7, 23, 23, Yes], [22, 8, 22, 22, Yes], [22, 9, 21, 21, Yes], [22, 10, 20, 20, Yes], [22, 11, 19, 19, Yes], [22, 12, 18, 18, Yes], [22, 13, 17, 17, Yes], [22, 14, 16, 16, Yes], [22, 15, 15, 15, Yes], [22, 16, 14, 14, Yes], [22, 17, 13, 13, Yes], [22, 18, 12, 12, Yes], [22, 19, 11, 11, Yes], [22, 20, 10, 10, Yes], [22, 21, 9, 9, Yes], [22, 22, 8, 8, Yes], [22, 23, 7, 7, Yes], [22, 24, 6, 6, Yes], [22, 25, 5, 5, Yes], [23, 0, 3, 3, Yes], [23, 1, 2, 2, Yes], [23, 2, 1, 1, Yes], [23, 3, 0, 0, Yes], [23, 4, 25, 25, Yes], [23, 5, 24, 24, Yes], [23, 6, 23, 23, Yes], [23, 7, 22, 22, Yes], [23, 8, 21, 21, Yes], [23, 9, 20, 20, Yes], [23, 10, 19, 19, Yes], [23, 11, 18, 18, Yes], [23, 12, 17, 17, Yes], [23, 13, 16, 16, Yes], [23, 14, 15, 15, Yes], [23, 15, 14, 14, Yes], [23, 16, 13, 13, Yes], [23, 17, 12, 12, Yes], [23, 18, 11, 11, Yes], [23, 19, 10, 10, Yes], [23, 20, 9, 9, Yes], [23, 21, 8, 8, Yes], [23, 22, 7, 7, Yes], [23, 23, 6, 6, Yes], [23, 24, 5, 5, Yes], [23, 25, 4, 4, Yes], [24, 0, 2, 2, Yes], [24, 1, 1, 1, Yes], [24, 2, 0, 0, Yes], [24, 3, 25, 25, Yes], [24, 4, 24, 24, Yes], [24, 5, 23, 23, Yes], [24, 6, 22, 22, Yes], [24, 7, 21, 21, Yes], [24, 8, 20, 20, Yes], [24, 9, 19, 19, Yes], [24, 10, 18, 18, Yes], [24, 11, 17, 17, Yes], [24, 12, 16, 16, Yes], [24, 13, 15, 15, Yes], [24, 14, 14, 14, Yes], [24, 15, 13, 13, Yes], [24, 16, 12, 12, Yes], [24, 17, 11, 11, Yes], [24, 18, 10, 10, Yes], [24, 19, 9, 9, Yes], [24, 20, 8, 8, Yes], [24, 21, 7, 7, Yes], [24, 22, 6, 6, Yes], [24, 23, 5, 5, Yes], [24, 24, 4, 4, Yes], [24, 25, 3, 3, Yes], [25, 0, 1, 1, Yes], [25, 1, 0, 0, Yes], [25, 2, 25, 25, Yes], [25, 3, 24, 24, Yes], [25, 4, 23, 23, Yes], [25, 5, 22, 22, Yes], [25, 6, 21, 21, Yes], [25, 7, 20, 20, Yes], [25, 8, 19, 19, Yes], [25, 9, 18, 18, Yes], [25, 10, 17, 17, Yes], [25, 11, 16, 16, Yes], [25, 12, 15, 15, Yes], [25, 13, 14, 14, Yes], [25, 14, 13, 13, Yes], [25, 15, 12, 12, Yes], [25, 16, 11, 11, Yes], [25, 17, 10, 10, Yes], [25, 18, 9, 9, Yes], [25, 19, 8, 8, Yes], [25, 20, 7, 7, Yes], [25, 21, 6, 6, Yes], [25, 22, 5, 5, Yes], [25, 23, 4, 4, Yes], [25, 24, 3, 3, Yes], [25, 25, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (26, 26, [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 25, 25, Yes], [0, 2, 24, 24, Yes], [0, 3, 23, 23, Yes], [0, 4, 22, 22, Yes], [0, 5, 21, 21, Yes], [0, 6, 20, 20, Yes], [0, 7, 19, 19, Yes], [0, 8, 18, 18, Yes], [0, 9, 17, 17, Yes], [0, 10, 16, 16, Yes], [0, 11, 15, 15, Yes], [0, 12, 14, 14, Yes], [0, 13, 13, 13, Yes], [0, 14, 12, 12, Yes], [0, 15, 11, 11, Yes], [0, 16, 10, 10, Yes], [0, 17, 9, 9, Yes], [0, 18, 8, 8, Yes], [0, 19, 7, 7, Yes], [0, 20, 6, 6, Yes], [0, 21, 5, 5, Yes], [0, 22, 4, 4, Yes], [0, 23, 3, 3, Yes], [0, 24, 2, 2, Yes], [0, 25, 1, 1, Yes], [1, 0, 25, 25, Yes], [1, 1, 24, 24, Yes], [1, 2, 23, 23, Yes], [1, 3, 22, 22, Yes], [1, 4, 21, 21, Yes], [1, 5, 20, 20, Yes], [1, 6, 19, 19, Yes], [1, 7, 18, 18, Yes], [1, 8, 17, 17, Yes], [1, 9, 16, 16, Yes], [1, 10, 15, 15, Yes], [1, 11, 14, 14, Yes], [1, 12, 13, 13, Yes], [1, 13, 12, 12, Yes], [1, 14, 11, 11, Yes], [1, 15, 10, 10, Yes], [1, 16, 9, 9, Yes], [1, 17, 8, 8, Yes], [1, 18, 7, 7, Yes], [1, 19, 6, 6, Yes], [1, 20, 5, 5, Yes], [1, 21, 4, 4, Yes], [1, 22, 3, 3, Yes], [1, 23, 2, 2, Yes], [1, 24, 1, 1, Yes], [1, 25, 0, 0, Yes], [2, 0, 24, 24, Yes], [2, 1, 23, 23, Yes], [2, 2, 22, 22, Yes], [2, 3, 21, 21, Yes], [2, 4, 20, 20, Yes], [2, 5, 19, 19, Yes], [2, 6, 18, 18, Yes], [2, 7, 17, 17, Yes], [2, 8, 16, 16, Yes], [2, 9, 15, 15, Yes], [2, 10, 14, 14, Yes], [2, 11, 13, 13, Yes], [2, 12, 12, 12, Yes], [2, 13, 11, 11, Yes], [2, 14, 10, 10, Yes], [2, 15, 9, 9, Yes], [2, 16, 8, 8, Yes], [2, 17, 7, 7, Yes], [2, 18, 6, 6, Yes], [2, 19, 5, 5, Yes], [2, 20, 4, 4, Yes], [2, 21, 3, 3, Yes], [2, 22, 2, 2, Yes], [2, 23, 1, 1, Yes], [2, 24, 0, 0, Yes], [2, 25, 25, 25, Yes], [3, 0, 23, 23, Yes], [3, 1, 22, 22, Yes], [3, 2, 21, 21, Yes], [3, 3, 20, 20, Yes], [3, 4, 19, 19, Yes], [3, 5, 18, 18, Yes], [3, 6, 17, 17, Yes], [3, 7, 16, 16, Yes], [3, 8, 15, 15, Yes], [3, 9, 14, 14, Yes], [3, 10, 13, 13, Yes], [3, 11, 12, 12, Yes], [3, 12, 11, 11, Yes], [3, 13, 10, 10, Yes], [3, 14, 9, 9, Yes], [3, 15, 8, 8, Yes], [3, 16, 7, 7, Yes], [3, 17, 6, 6, Yes], [3, 18, 5, 5, Yes], [3, 19, 4, 4, Yes], [3, 20, 3, 3, Yes], [3, 21, 2, 2, Yes], [3, 22, 1, 1, Yes], [3, 23, 0, 0, Yes], [3, 24, 25, 25, Yes], [3, 25, 24, 24, Yes], [4, 0, 22, 22, Yes], [4, 1, 21, 21, Yes], [4, 2, 20, 20, Yes], [4, 3, 19, 19, Yes], [4, 4, 18, 18, Yes], [4, 5, 17, 17, Yes], [4, 6, 16, 16, Yes], [4, 7, 15, 15, Yes], [4, 8, 14, 14, Yes], [4, 9, 13, 13, Yes], [4, 10, 12, 12, Yes], [4, 11, 11, 11, Yes], [4, 12, 10, 10, Yes], [4, 13, 9, 9, Yes], [4, 14, 8, 8, Yes], [4, 15, 7, 7, Yes], [4, 16, 6, 6, Yes], [4, 17, 5, 5, Yes], [4, 18, 4, 4, Yes], [4, 19, 3, 3, Yes], [4, 20, 2, 2, Yes], [4, 21, 1, 1, Yes], [4, 22, 0, 0, Yes], [4, 23, 25, 25, Yes], [4, 24, 24, 24, Yes], [4, 25, 23, 23, Yes], [5, 0, 21, 21, Yes], [5, 1, 20, 20, Yes], [5, 2, 19, 19, Yes], [5, 3, 18, 18, Yes], [5, 4, 17, 17, Yes], [5, 5, 16, 16, Yes], [5, 6, 15, 15, Yes], [5, 7, 14, 14, Yes], [5, 8, 13, 13, Yes], [5, 9, 12, 12, Yes], [5, 10, 11, 11, Yes], [5, 11, 10, 10, Yes], [5, 12, 9, 9, Yes], [5, 13, 8, 8, Yes], [5, 14, 7, 7, Yes], [5, 15, 6, 6, Yes], [5, 16, 5, 5, Yes], [5, 17, 4, 4, Yes], [5, 18, 3, 3, Yes], [5, 19, 2, 2, Yes], [5, 20, 1, 1, Yes], [5, 21, 0, 0, Yes], [5, 22, 25, 25, Yes], [5, 23, 24, 24, Yes], [5, 24, 23, 23, Yes], [5, 25, 22, 22, Yes], [6, 0, 20, 20, Yes], [6, 1, 19, 19, Yes], [6, 2, 18, 18, Yes], [6, 3, 17, 17, Yes], [6, 4, 16, 16, Yes], [6, 5, 15, 15, Yes], [6, 6, 14, 14, Yes], [6, 7, 13, 13, Yes], [6, 8, 12, 12, Yes], [6, 9, 11, 11, Yes], [6, 10, 10, 10, Yes], [6, 11, 9, 9, Yes], [6, 12, 8, 8, Yes], [6, 13, 7, 7, Yes], [6, 14, 6, 6, Yes], [6, 15, 5, 5, Yes], [6, 16, 4, 4, Yes], [6, 17, 3, 3, Yes], [6, 18, 2, 2, Yes], [6, 19, 1, 1, Yes], [6, 20, 0, 0, Yes], [6, 21, 25, 25, Yes], [6, 22, 24, 24, Yes], [6, 23, 23, 23, Yes], [6, 24, 22, 22, Yes], [6, 25, 21, 21, Yes], [7, 0, 19, 19, Yes], [7, 1, 18, 18, Yes], [7, 2, 17, 17, Yes], [7, 3, 16, 16, Yes], [7, 4, 15, 15, Yes], [7, 5, 14, 14, Yes], [7, 6, 13, 13, Yes], [7, 7, 12, 12, Yes], [7, 8, 11, 11, Yes], [7, 9, 10, 10, Yes], [7, 10, 9, 9, Yes], [7, 11, 8, 8, Yes], [7, 12, 7, 7, Yes], [7, 13, 6, 6, Yes], [7, 14, 5, 5, Yes], [7, 15, 4, 4, Yes], [7, 16, 3, 3, Yes], [7, 17, 2, 2, Yes], [7, 18, 1, 1, Yes], [7, 19, 0, 0, Yes], [7, 20, 25, 25, Yes], [7, 21, 24, 24, Yes], [7, 22, 23, 23, Yes], [7, 23, 22, 22, Yes], [7, 24, 21, 21, Yes], [7, 25, 20, 20, Yes], [8, 0, 18, 18, Yes], [8, 1, 17, 17, Yes], [8, 2, 16, 16, Yes], [8, 3, 15, 15, Yes], [8, 4, 14, 14, Yes], [8, 5, 13, 13, Yes], [8, 6, 12, 12, Yes], [8, 7, 11, 11, Yes], [8, 8, 10, 10, Yes], [8, 9, 9, 9, Yes], [8, 10, 8, 8, Yes], [8, 11, 7, 7, Yes], [8, 12, 6, 6, Yes], [8, 13, 5, 5, Yes], [8, 14, 4, 4, Yes], [8, 15, 3, 3, Yes], [8, 16, 2, 2, Yes], [8, 17, 1, 1, Yes], [8, 18, 0, 0, Yes], [8, 19, 25, 25, Yes], [8, 20, 24, 24, Yes], [8, 21, 23, 23, Yes], [8, 22, 22, 22, Yes], [8, 23, 21, 21, Yes], [8, 24, 20, 20, Yes], [8, 25, 19, 19, Yes], [9, 0, 17, 17, Yes], [9, 1, 16, 16, Yes], [9, 2, 15, 15, Yes], [9, 3, 14, 14, Yes], [9, 4, 13, 13, Yes], [9, 5, 12, 12, Yes], [9, 6, 11, 11, Yes], [9, 7, 10, 10, Yes], [9, 8, 9, 9, Yes], [9, 9, 8, 8, Yes], [9, 10, 7, 7, Yes], [9, 11, 6, 6, Yes], [9, 12, 5, 5, Yes], [9, 13, 4, 4, Yes], [9, 14, 3, 3, Yes], [9, 15, 2, 2, Yes], [9, 16, 1, 1, Yes], [9, 17, 0, 0, Yes], [9, 18, 25, 25, Yes], [9, 19, 24, 24, Yes], [9, 20, 23, 23, Yes], [9, 21, 22, 22, Yes], [9, 22, 21, 21, Yes], [9, 23, 20, 20, Yes], [9, 24, 19, 19, Yes], [9, 25, 18, 18, Yes], [10, 0, 16, 16, Yes], [10, 1, 15, 15, Yes], [10, 2, 14, 14, Yes], [10, 3, 13, 13, Yes], [10, 4, 12, 12, Yes], [10, 5, 11, 11, Yes], [10, 6, 10, 10, Yes], [10, 7, 9, 9, Yes], [10, 8, 8, 8, Yes], [10, 9, 7, 7, Yes], [10, 10, 6, 6, Yes], [10, 11, 5, 5, Yes], [10, 12, 4, 4, Yes], [10, 13, 3, 3, Yes], [10, 14, 2, 2, Yes], [10, 15, 1, 1, Yes], [10, 16, 0, 0, Yes], [10, 17, 25, 25, Yes], [10, 18, 24, 24, Yes], [10, 19, 23, 23, Yes], [10, 20, 22, 22, Yes], [10, 21, 21, 21, Yes], [10, 22, 20, 20, Yes], [10, 23, 19, 19, Yes], [10, 24, 18, 18, Yes], [10, 25, 17, 17, Yes], [11, 0, 15, 15, Yes], [11, 1, 14, 14, Yes], [11, 2, 13, 13, Yes], [11, 3, 12, 12, Yes], [11, 4, 11, 11, Yes], [11, 5, 10, 10, Yes], [11, 6, 9, 9, Yes], [11, 7, 8, 8, Yes], [11, 8, 7, 7, Yes], [11, 9, 6, 6, Yes], [11, 10, 5, 5, Yes], [11, 11, 4, 4, Yes], [11, 12, 3, 3, Yes], [11, 13, 2, 2, Yes], [11, 14, 1, 1, Yes], [11, 15, 0, 0, Yes], [11, 16, 25, 25, Yes], [11, 17, 24, 24, Yes], [11, 18, 23, 23, Yes], [11, 19, 22, 22, Yes], [11, 20, 21, 21, Yes], [11, 21, 20, 20, Yes], [11, 22, 19, 19, Yes], [11, 23, 18, 18, Yes], [11, 24, 17, 17, Yes], [11, 25, 16, 16, Yes], [12, 0, 14, 14, Yes], [12, 1, 13, 13, Yes], [12, 2, 12, 12, Yes], [12, 3, 11, 11, Yes], [12, 4, 10, 10, Yes], [12, 5, 9, 9, Yes], [12, 6, 8, 8, Yes], [12, 7, 7, 7, Yes], [12, 8, 6, 6, Yes], [12, 9, 5, 5, Yes], [12, 10, 4, 4, Yes], [12, 11, 3, 3, Yes], [12, 12, 2, 2, Yes], [12, 13, 1, 1, Yes], [12, 14, 0, 0, Yes], [12, 15, 25, 25, Yes], [12, 16, 24, 24, Yes], [12, 17, 23, 23, Yes], [12, 18, 22, 22, Yes], [12, 19, 21, 21, Yes], [12, 20, 20, 20, Yes], [12, 21, 19, 19, Yes], [12, 22, 18, 18, Yes], [12, 23, 17, 17, Yes], [12, 24, 16, 16, Yes], [12, 25, 15, 15, Yes], [13, 0, 13, 13, Yes], [13, 1, 12, 12, Yes], [13, 2, 11, 11, Yes], [13, 3, 10, 10, Yes], [13, 4, 9, 9, Yes], [13, 5, 8, 8, Yes], [13, 6, 7, 7, Yes], [13, 7, 6, 6, Yes], [13, 8, 5, 5, Yes], [13, 9, 4, 4, Yes], [13, 10, 3, 3, Yes], [13, 11, 2, 2, Yes], [13, 12, 1, 1, Yes], [13, 13, 0, 0, Yes], [13, 14, 25, 25, Yes], [13, 15, 24, 24, Yes], [13, 16, 23, 23, Yes], [13, 17, 22, 22, Yes], [13, 18, 21, 21, Yes], [13, 19, 20, 20, Yes], [13, 20, 19, 19, Yes], [13, 21, 18, 18, Yes], [13, 22, 17, 17, Yes], [13, 23, 16, 16, Yes], [13, 24, 15, 15, Yes], [13, 25, 14, 14, Yes], [14, 0, 12, 12, Yes], [14, 1, 11, 11, Yes], [14, 2, 10, 10, Yes], [14, 3, 9, 9, Yes], [14, 4, 8, 8, Yes], [14, 5, 7, 7, Yes], [14, 6, 6, 6, Yes], [14, 7, 5, 5, Yes], [14, 8, 4, 4, Yes], [14, 9, 3, 3, Yes], [14, 10, 2, 2, Yes], [14, 11, 1, 1, Yes], [14, 12, 0, 0, Yes], [14, 13, 25, 25, Yes], [14, 14, 24, 24, Yes], [14, 15, 23, 23, Yes], [14, 16, 22, 22, Yes], [14, 17, 21, 21, Yes], [14, 18, 20, 20, Yes], [14, 19, 19, 19, Yes], [14, 20, 18, 18, Yes], [14, 21, 17, 17, Yes], [14, 22, 16, 16, Yes], [14, 23, 15, 15, Yes], [14, 24, 14, 14, Yes], [14, 25, 13, 13, Yes], [15, 0, 11, 11, Yes], [15, 1, 10, 10, Yes], [15, 2, 9, 9, Yes], [15, 3, 8, 8, Yes], [15, 4, 7, 7, Yes], [15, 5, 6, 6, Yes], [15, 6, 5, 5, Yes], [15, 7, 4, 4, Yes], [15, 8, 3, 3, Yes], [15, 9, 2, 2, Yes], [15, 10, 1, 1, Yes], [15, 11, 0, 0, Yes], [15, 12, 25, 25, Yes], [15, 13, 24, 24, Yes], [15, 14, 23, 23, Yes], [15, 15, 22, 22, Yes], [15, 16, 21, 21, Yes], [15, 17, 20, 20, Yes], [15, 18, 19, 19, Yes], [15, 19, 18, 18, Yes], [15, 20, 17, 17, Yes], [15, 21, 16, 16, Yes], [15, 22, 15, 15, Yes], [15, 23, 14, 14, Yes], [15, 24, 13, 13, Yes], [15, 25, 12, 12, Yes], [16, 0, 10, 10, Yes], [16, 1, 9, 9, Yes], [16, 2, 8, 8, Yes], [16, 3, 7, 7, Yes], [16, 4, 6, 6, Yes], [16, 5, 5, 5, Yes], [16, 6, 4, 4, Yes], [16, 7, 3, 3, Yes], [16, 8, 2, 2, Yes], [16, 9, 1, 1, Yes], [16, 10, 0, 0, Yes], [16, 11, 25, 25, Yes], [16, 12, 24, 24, Yes], [16, 13, 23, 23, Yes], [16, 14, 22, 22, Yes], [16, 15, 21, 21, Yes], [16, 16, 20, 20, Yes], [16, 17, 19, 19, Yes], [16, 18, 18, 18, Yes], [16, 19, 17, 17, Yes], [16, 20, 16, 16, Yes], [16, 21, 15, 15, Yes], [16, 22, 14, 14, Yes], [16, 23, 13, 13, Yes], [16, 24, 12, 12, Yes], [16, 25, 11, 11, Yes], [17, 0, 9, 9, Yes], [17, 1, 8, 8, Yes], [17, 2, 7, 7, Yes], [17, 3, 6, 6, Yes], [17, 4, 5, 5, Yes], [17, 5, 4, 4, Yes], [17, 6, 3, 3, Yes], [17, 7, 2, 2, Yes], [17, 8, 1, 1, Yes], [17, 9, 0, 0, Yes], [17, 10, 25, 25, Yes], [17, 11, 24, 24, Yes], [17, 12, 23, 23, Yes], [17, 13, 22, 22, Yes], [17, 14, 21, 21, Yes], [17, 15, 20, 20, Yes], [17, 16, 19, 19, Yes], [17, 17, 18, 18, Yes], [17, 18, 17, 17, Yes], [17, 19, 16, 16, Yes], [17, 20, 15, 15, Yes], [17, 21, 14, 14, Yes], [17, 22, 13, 13, Yes], [17, 23, 12, 12, Yes], [17, 24, 11, 11, Yes], [17, 25, 10, 10, Yes], [18, 0, 8, 8, Yes], [18, 1, 7, 7, Yes], [18, 2, 6, 6, Yes], [18, 3, 5, 5, Yes], [18, 4, 4, 4, Yes], [18, 5, 3, 3, Yes], [18, 6, 2, 2, Yes], [18, 7, 1, 1, Yes], [18, 8, 0, 0, Yes], [18, 9, 25, 25, Yes], [18, 10, 24, 24, Yes], [18, 11, 23, 23, Yes], [18, 12, 22, 22, Yes], [18, 13, 21, 21, Yes], [18, 14, 20, 20, Yes], [18, 15, 19, 19, Yes], [18, 16, 18, 18, Yes], [18, 17, 17, 17, Yes], [18, 18, 16, 16, Yes], [18, 19, 15, 15, Yes], [18, 20, 14, 14, Yes], [18, 21, 13, 13, Yes], [18, 22, 12, 12, Yes], [18, 23, 11, 11, Yes], [18, 24, 10, 10, Yes], [18, 25, 9, 9, Yes], [19, 0, 7, 7, Yes], [19, 1, 6, 6, Yes], [19, 2, 5, 5, Yes], [19, 3, 4, 4, Yes], [19, 4, 3, 3, Yes], [19, 5, 2, 2, Yes], [19, 6, 1, 1, Yes], [19, 7, 0, 0, Yes], [19, 8, 25, 25, Yes], [19, 9, 24, 24, Yes], [19, 10, 23, 23, Yes], [19, 11, 22, 22, Yes], [19, 12, 21, 21, Yes], [19, 13, 20, 20, Yes], [19, 14, 19, 19, Yes], [19, 15, 18, 18, Yes], [19, 16, 17, 17, Yes], [19, 17, 16, 16, Yes], [19, 18, 15, 15, Yes], [19, 19, 14, 14, Yes], [19, 20, 13, 13, Yes], [19, 21, 12, 12, Yes], [19, 22, 11, 11, Yes], [19, 23, 10, 10, Yes], [19, 24, 9, 9, Yes], [19, 25, 8, 8, Yes], [20, 0, 6, 6, Yes], [20, 1, 5, 5, Yes], [20, 2, 4, 4, Yes], [20, 3, 3, 3, Yes], [20, 4, 2, 2, Yes], [20, 5, 1, 1, Yes], [20, 6, 0, 0, Yes], [20, 7, 25, 25, Yes], [20, 8, 24, 24, Yes], [20, 9, 23, 23, Yes], [20, 10, 22, 22, Yes], [20, 11, 21, 21, Yes], [20, 12, 20, 20, Yes], [20, 13, 19, 19, Yes], [20, 14, 18, 18, Yes], [20, 15, 17, 17, Yes], [20, 16, 16, 16, Yes], [20, 17, 15, 15, Yes], [20, 18, 14, 14, Yes], [20, 19, 13, 13, Yes], [20, 20, 12, 12, Yes], [20, 21, 11, 11, Yes], [20, 22, 10, 10, Yes], [20, 23, 9, 9, Yes], [20, 24, 8, 8, Yes], [20, 25, 7, 7, Yes], [21, 0, 5, 5, Yes], [21, 1, 4, 4, Yes], [21, 2, 3, 3, Yes], [21, 3, 2, 2, Yes], [21, 4, 1, 1, Yes], [21, 5, 0, 0, Yes], [21, 6, 25, 25, Yes], [21, 7, 24, 24, Yes], [21, 8, 23, 23, Yes], [21, 9, 22, 22, Yes], [21, 10, 21, 21, Yes], [21, 11, 20, 20, Yes], [21, 12, 19, 19, Yes], [21, 13, 18, 18, Yes], [21, 14, 17, 17, Yes], [21, 15, 16, 16, Yes], [21, 16, 15, 15, Yes], [21, 17, 14, 14, Yes], [21, 18, 13, 13, Yes], [21, 19, 12, 12, Yes], [21, 20, 11, 11, Yes], [21, 21, 10, 10, Yes], [21, 22, 9, 9, Yes], [21, 23, 8, 8, Yes], [21, 24, 7, 7, Yes], [21, 25, 6, 6, Yes], [22, 0, 4, 4, Yes], [22, 1, 3, 3, Yes], [22, 2, 2, 2, Yes], [22, 3, 1, 1, Yes], [22, 4, 0, 0, Yes], [22, 5, 25, 25, Yes], [22, 6, 24, 24, Yes], [22, 7, 23, 23, Yes], [22, 8, 22, 22, Yes], [22, 9, 21, 21, Yes], [22, 10, 20, 20, Yes], [22, 11, 19, 19, Yes], [22, 12, 18, 18, Yes], [22, 13, 17, 17, Yes], [22, 14, 16, 16, Yes], [22, 15, 15, 15, Yes], [22, 16, 14, 14, Yes], [22, 17, 13, 13, Yes], [22, 18, 12, 12, Yes], [22, 19, 11, 11, Yes], [22, 20, 10, 10, Yes], [22, 21, 9, 9, Yes], [22, 22, 8, 8, Yes], [22, 23, 7, 7, Yes], [22, 24, 6, 6, Yes], [22, 25, 5, 5, Yes], [23, 0, 3, 3, Yes], [23, 1, 2, 2, Yes], [23, 2, 1, 1, Yes], [23, 3, 0, 0, Yes], [23, 4, 25, 25, Yes], [23, 5, 24, 24, Yes], [23, 6, 23, 23, Yes], [23, 7, 22, 22, Yes], [23, 8, 21, 21, Yes], [23, 9, 20, 20, Yes], [23, 10, 19, 19, Yes], [23, 11, 18, 18, Yes], [23, 12, 17, 17, Yes], [23, 13, 16, 16, Yes], [23, 14, 15, 15, Yes], [23, 15, 14, 14, Yes], [23, 16, 13, 13, Yes], [23, 17, 12, 12, Yes], [23, 18, 11, 11, Yes], [23, 19, 10, 10, Yes], [23, 20, 9, 9, Yes], [23, 21, 8, 8, Yes], [23, 22, 7, 7, Yes], [23, 23, 6, 6, Yes], [23, 24, 5, 5, Yes], [23, 25, 4, 4, Yes], [24, 0, 2, 2, Yes], [24, 1, 1, 1, Yes], [24, 2, 0, 0, Yes], [24, 3, 25, 25, Yes], [24, 4, 24, 24, Yes], [24, 5, 23, 23, Yes], [24, 6, 22, 22, Yes], [24, 7, 21, 21, Yes], [24, 8, 20, 20, Yes], [24, 9, 19, 19, Yes], [24, 10, 18, 18, Yes], [24, 11, 17, 17, Yes], [24, 12, 16, 16, Yes], [24, 13, 15, 15, Yes], [24, 14, 14, 14, Yes], [24, 15, 13, 13, Yes], [24, 16, 12, 12, Yes], [24, 17, 11, 11, Yes], [24, 18, 10, 10, Yes], [24, 19, 9, 9, Yes], [24, 20, 8, 8, Yes], [24, 21, 7, 7, Yes], [24, 22, 6, 6, Yes], [24, 23, 5, 5, Yes], [24, 24, 4, 4, Yes], [24, 25, 3, 3, Yes], [25, 0, 1, 1, Yes], [25, 1, 0, 0, Yes], [25, 2, 25, 25, Yes], [25, 3, 24, 24, Yes], [25, 4, 23, 23, Yes], [25, 5, 22, 22, Yes], [25, 6, 21, 21, Yes], [25, 7, 20, 20, Yes], [25, 8, 19, 19, Yes], [25, 9, 18, 18, Yes], [25, 10, 17, 17, Yes], [25, 11, 16, 16, Yes], [25, 12, 15, 15, Yes], [25, 13, 14, 14, Yes], [25, 14, 13, 13, Yes], [25, 15, 12, 12, Yes], [25, 16, 11, 11, Yes], [25, 17, 10, 10, Yes], [25, 18, 9, 9, Yes], [25, 19, 8, 8, Yes], [25, 20, 7, 7, Yes], [25, 21, 6, 6, Yes], [25, 22, 5, 5, Yes], [25, 23, 4, 4, Yes], [25, 24, 3, 3, Yes], [25, 25, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0025","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c059dbd92d1f6dee455bcae62fd2387494c5b359eb2fe715d4bf93b98f9da67","family_id":"qf-mod45-kernel-homomorphism-qualification","seed":202611141025,"semantic_template":"qf-mod45-kernel-homomorphism-qualification/{\"mapping\":[0,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]]}"},"fields":{"title":"Qualify Both Groups and Every Product Pair before Applying Kernel Theorems","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], target identity=0, and total mapping=[0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).","answer":{"kind":"expression","value":"(27, 27, [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 26, 26, Yes], [0, 2, 25, 25, Yes], [0, 3, 24, 24, Yes], [0, 4, 23, 23, Yes], [0, 5, 22, 22, Yes], [0, 6, 21, 21, Yes], [0, 7, 20, 20, Yes], [0, 8, 19, 19, Yes], [0, 9, 18, 18, Yes], [0, 10, 17, 17, Yes], [0, 11, 16, 16, Yes], [0, 12, 15, 15, Yes], [0, 13, 14, 14, Yes], [0, 14, 13, 13, Yes], [0, 15, 12, 12, Yes], [0, 16, 11, 11, Yes], [0, 17, 10, 10, Yes], [0, 18, 9, 9, Yes], [0, 19, 8, 8, Yes], [0, 20, 7, 7, Yes], [0, 21, 6, 6, Yes], [0, 22, 5, 5, Yes], [0, 23, 4, 4, Yes], [0, 24, 3, 3, Yes], [0, 25, 2, 2, Yes], [0, 26, 1, 1, Yes], [1, 0, 26, 26, Yes], [1, 1, 25, 25, Yes], [1, 2, 24, 24, Yes], [1, 3, 23, 23, Yes], [1, 4, 22, 22, Yes], [1, 5, 21, 21, Yes], [1, 6, 20, 20, Yes], [1, 7, 19, 19, Yes], [1, 8, 18, 18, Yes], [1, 9, 17, 17, Yes], [1, 10, 16, 16, Yes], [1, 11, 15, 15, Yes], [1, 12, 14, 14, Yes], [1, 13, 13, 13, Yes], [1, 14, 12, 12, Yes], [1, 15, 11, 11, Yes], [1, 16, 10, 10, Yes], [1, 17, 9, 9, Yes], [1, 18, 8, 8, Yes], [1, 19, 7, 7, Yes], [1, 20, 6, 6, Yes], [1, 21, 5, 5, Yes], [1, 22, 4, 4, Yes], [1, 23, 3, 3, Yes], [1, 24, 2, 2, Yes], [1, 25, 1, 1, Yes], [1, 26, 0, 0, Yes], [2, 0, 25, 25, Yes], [2, 1, 24, 24, Yes], [2, 2, 23, 23, Yes], [2, 3, 22, 22, Yes], [2, 4, 21, 21, Yes], [2, 5, 20, 20, Yes], [2, 6, 19, 19, Yes], [2, 7, 18, 18, Yes], [2, 8, 17, 17, Yes], [2, 9, 16, 16, Yes], [2, 10, 15, 15, Yes], [2, 11, 14, 14, Yes], [2, 12, 13, 13, Yes], [2, 13, 12, 12, Yes], [2, 14, 11, 11, Yes], [2, 15, 10, 10, Yes], [2, 16, 9, 9, Yes], [2, 17, 8, 8, Yes], [2, 18, 7, 7, Yes], [2, 19, 6, 6, Yes], [2, 20, 5, 5, Yes], [2, 21, 4, 4, Yes], [2, 22, 3, 3, Yes], [2, 23, 2, 2, Yes], [2, 24, 1, 1, Yes], [2, 25, 0, 0, Yes], [2, 26, 26, 26, Yes], [3, 0, 24, 24, Yes], [3, 1, 23, 23, Yes], [3, 2, 22, 22, Yes], [3, 3, 21, 21, Yes], [3, 4, 20, 20, Yes], [3, 5, 19, 19, Yes], [3, 6, 18, 18, Yes], [3, 7, 17, 17, Yes], [3, 8, 16, 16, Yes], [3, 9, 15, 15, Yes], [3, 10, 14, 14, Yes], [3, 11, 13, 13, Yes], [3, 12, 12, 12, Yes], [3, 13, 11, 11, Yes], [3, 14, 10, 10, Yes], [3, 15, 9, 9, Yes], [3, 16, 8, 8, Yes], [3, 17, 7, 7, Yes], [3, 18, 6, 6, Yes], [3, 19, 5, 5, Yes], [3, 20, 4, 4, Yes], [3, 21, 3, 3, Yes], [3, 22, 2, 2, Yes], [3, 23, 1, 1, Yes], [3, 24, 0, 0, Yes], [3, 25, 26, 26, Yes], [3, 26, 25, 25, Yes], [4, 0, 23, 23, Yes], [4, 1, 22, 22, Yes], [4, 2, 21, 21, Yes], [4, 3, 20, 20, Yes], [4, 4, 19, 19, Yes], [4, 5, 18, 18, Yes], [4, 6, 17, 17, Yes], [4, 7, 16, 16, Yes], [4, 8, 15, 15, Yes], [4, 9, 14, 14, Yes], [4, 10, 13, 13, Yes], [4, 11, 12, 12, Yes], [4, 12, 11, 11, Yes], [4, 13, 10, 10, Yes], [4, 14, 9, 9, Yes], [4, 15, 8, 8, Yes], [4, 16, 7, 7, Yes], [4, 17, 6, 6, Yes], [4, 18, 5, 5, Yes], [4, 19, 4, 4, Yes], [4, 20, 3, 3, Yes], [4, 21, 2, 2, Yes], [4, 22, 1, 1, Yes], [4, 23, 0, 0, Yes], [4, 24, 26, 26, Yes], [4, 25, 25, 25, Yes], [4, 26, 24, 24, Yes], [5, 0, 22, 22, Yes], [5, 1, 21, 21, Yes], [5, 2, 20, 20, Yes], [5, 3, 19, 19, Yes], [5, 4, 18, 18, Yes], [5, 5, 17, 17, Yes], [5, 6, 16, 16, Yes], [5, 7, 15, 15, Yes], [5, 8, 14, 14, Yes], [5, 9, 13, 13, Yes], [5, 10, 12, 12, Yes], [5, 11, 11, 11, Yes], [5, 12, 10, 10, Yes], [5, 13, 9, 9, Yes], [5, 14, 8, 8, Yes], [5, 15, 7, 7, Yes], [5, 16, 6, 6, Yes], [5, 17, 5, 5, Yes], [5, 18, 4, 4, Yes], [5, 19, 3, 3, Yes], [5, 20, 2, 2, Yes], [5, 21, 1, 1, Yes], [5, 22, 0, 0, Yes], [5, 23, 26, 26, Yes], [5, 24, 25, 25, Yes], [5, 25, 24, 24, Yes], [5, 26, 23, 23, Yes], [6, 0, 21, 21, Yes], [6, 1, 20, 20, Yes], [6, 2, 19, 19, Yes], [6, 3, 18, 18, Yes], [6, 4, 17, 17, Yes], [6, 5, 16, 16, Yes], [6, 6, 15, 15, Yes], [6, 7, 14, 14, Yes], [6, 8, 13, 13, Yes], [6, 9, 12, 12, Yes], [6, 10, 11, 11, Yes], [6, 11, 10, 10, Yes], [6, 12, 9, 9, Yes], [6, 13, 8, 8, Yes], [6, 14, 7, 7, Yes], [6, 15, 6, 6, Yes], [6, 16, 5, 5, Yes], [6, 17, 4, 4, Yes], [6, 18, 3, 3, Yes], [6, 19, 2, 2, Yes], [6, 20, 1, 1, Yes], [6, 21, 0, 0, Yes], [6, 22, 26, 26, Yes], [6, 23, 25, 25, Yes], [6, 24, 24, 24, Yes], [6, 25, 23, 23, Yes], [6, 26, 22, 22, Yes], [7, 0, 20, 20, Yes], [7, 1, 19, 19, Yes], [7, 2, 18, 18, Yes], [7, 3, 17, 17, Yes], [7, 4, 16, 16, Yes], [7, 5, 15, 15, Yes], [7, 6, 14, 14, Yes], [7, 7, 13, 13, Yes], [7, 8, 12, 12, Yes], [7, 9, 11, 11, Yes], [7, 10, 10, 10, Yes], [7, 11, 9, 9, Yes], [7, 12, 8, 8, Yes], [7, 13, 7, 7, Yes], [7, 14, 6, 6, Yes], [7, 15, 5, 5, Yes], [7, 16, 4, 4, Yes], [7, 17, 3, 3, Yes], [7, 18, 2, 2, Yes], [7, 19, 1, 1, Yes], [7, 20, 0, 0, Yes], [7, 21, 26, 26, Yes], [7, 22, 25, 25, Yes], [7, 23, 24, 24, Yes], [7, 24, 23, 23, Yes], [7, 25, 22, 22, Yes], [7, 26, 21, 21, Yes], [8, 0, 19, 19, Yes], [8, 1, 18, 18, Yes], [8, 2, 17, 17, Yes], [8, 3, 16, 16, Yes], [8, 4, 15, 15, Yes], [8, 5, 14, 14, Yes], [8, 6, 13, 13, Yes], [8, 7, 12, 12, Yes], [8, 8, 11, 11, Yes], [8, 9, 10, 10, Yes], [8, 10, 9, 9, Yes], [8, 11, 8, 8, Yes], [8, 12, 7, 7, Yes], [8, 13, 6, 6, Yes], [8, 14, 5, 5, Yes], [8, 15, 4, 4, Yes], [8, 16, 3, 3, Yes], [8, 17, 2, 2, Yes], [8, 18, 1, 1, Yes], [8, 19, 0, 0, Yes], [8, 20, 26, 26, Yes], [8, 21, 25, 25, Yes], [8, 22, 24, 24, Yes], [8, 23, 23, 23, Yes], [8, 24, 22, 22, Yes], [8, 25, 21, 21, Yes], [8, 26, 20, 20, Yes], [9, 0, 18, 18, Yes], [9, 1, 17, 17, Yes], [9, 2, 16, 16, Yes], [9, 3, 15, 15, Yes], [9, 4, 14, 14, Yes], [9, 5, 13, 13, Yes], [9, 6, 12, 12, Yes], [9, 7, 11, 11, Yes], [9, 8, 10, 10, Yes], [9, 9, 9, 9, Yes], [9, 10, 8, 8, Yes], [9, 11, 7, 7, Yes], [9, 12, 6, 6, Yes], [9, 13, 5, 5, Yes], [9, 14, 4, 4, Yes], [9, 15, 3, 3, Yes], [9, 16, 2, 2, Yes], [9, 17, 1, 1, Yes], [9, 18, 0, 0, Yes], [9, 19, 26, 26, Yes], [9, 20, 25, 25, Yes], [9, 21, 24, 24, Yes], [9, 22, 23, 23, Yes], [9, 23, 22, 22, Yes], [9, 24, 21, 21, Yes], [9, 25, 20, 20, Yes], [9, 26, 19, 19, Yes], [10, 0, 17, 17, Yes], [10, 1, 16, 16, Yes], [10, 2, 15, 15, Yes], [10, 3, 14, 14, Yes], [10, 4, 13, 13, Yes], [10, 5, 12, 12, Yes], [10, 6, 11, 11, Yes], [10, 7, 10, 10, Yes], [10, 8, 9, 9, Yes], [10, 9, 8, 8, Yes], [10, 10, 7, 7, Yes], [10, 11, 6, 6, Yes], [10, 12, 5, 5, Yes], [10, 13, 4, 4, Yes], [10, 14, 3, 3, Yes], [10, 15, 2, 2, Yes], [10, 16, 1, 1, Yes], [10, 17, 0, 0, Yes], [10, 18, 26, 26, Yes], [10, 19, 25, 25, Yes], [10, 20, 24, 24, Yes], [10, 21, 23, 23, Yes], [10, 22, 22, 22, Yes], [10, 23, 21, 21, Yes], [10, 24, 20, 20, Yes], [10, 25, 19, 19, Yes], [10, 26, 18, 18, Yes], [11, 0, 16, 16, Yes], [11, 1, 15, 15, Yes], [11, 2, 14, 14, Yes], [11, 3, 13, 13, Yes], [11, 4, 12, 12, Yes], [11, 5, 11, 11, Yes], [11, 6, 10, 10, Yes], [11, 7, 9, 9, Yes], [11, 8, 8, 8, Yes], [11, 9, 7, 7, Yes], [11, 10, 6, 6, Yes], [11, 11, 5, 5, Yes], [11, 12, 4, 4, Yes], [11, 13, 3, 3, Yes], [11, 14, 2, 2, Yes], [11, 15, 1, 1, Yes], [11, 16, 0, 0, Yes], [11, 17, 26, 26, Yes], [11, 18, 25, 25, Yes], [11, 19, 24, 24, Yes], [11, 20, 23, 23, Yes], [11, 21, 22, 22, Yes], [11, 22, 21, 21, Yes], [11, 23, 20, 20, Yes], [11, 24, 19, 19, Yes], [11, 25, 18, 18, Yes], [11, 26, 17, 17, Yes], [12, 0, 15, 15, Yes], [12, 1, 14, 14, Yes], [12, 2, 13, 13, Yes], [12, 3, 12, 12, Yes], [12, 4, 11, 11, Yes], [12, 5, 10, 10, Yes], [12, 6, 9, 9, Yes], [12, 7, 8, 8, Yes], [12, 8, 7, 7, Yes], [12, 9, 6, 6, Yes], [12, 10, 5, 5, Yes], [12, 11, 4, 4, Yes], [12, 12, 3, 3, Yes], [12, 13, 2, 2, Yes], [12, 14, 1, 1, Yes], [12, 15, 0, 0, Yes], [12, 16, 26, 26, Yes], [12, 17, 25, 25, Yes], [12, 18, 24, 24, Yes], [12, 19, 23, 23, Yes], [12, 20, 22, 22, Yes], [12, 21, 21, 21, Yes], [12, 22, 20, 20, Yes], [12, 23, 19, 19, Yes], [12, 24, 18, 18, Yes], [12, 25, 17, 17, Yes], [12, 26, 16, 16, Yes], [13, 0, 14, 14, Yes], [13, 1, 13, 13, Yes], [13, 2, 12, 12, Yes], [13, 3, 11, 11, Yes], [13, 4, 10, 10, Yes], [13, 5, 9, 9, Yes], [13, 6, 8, 8, Yes], [13, 7, 7, 7, Yes], [13, 8, 6, 6, Yes], [13, 9, 5, 5, Yes], [13, 10, 4, 4, Yes], [13, 11, 3, 3, Yes], [13, 12, 2, 2, Yes], [13, 13, 1, 1, Yes], [13, 14, 0, 0, Yes], [13, 15, 26, 26, Yes], [13, 16, 25, 25, Yes], [13, 17, 24, 24, Yes], [13, 18, 23, 23, Yes], [13, 19, 22, 22, Yes], [13, 20, 21, 21, Yes], [13, 21, 20, 20, Yes], [13, 22, 19, 19, Yes], [13, 23, 18, 18, Yes], [13, 24, 17, 17, Yes], [13, 25, 16, 16, Yes], [13, 26, 15, 15, Yes], [14, 0, 13, 13, Yes], [14, 1, 12, 12, Yes], [14, 2, 11, 11, Yes], [14, 3, 10, 10, Yes], [14, 4, 9, 9, Yes], [14, 5, 8, 8, Yes], [14, 6, 7, 7, Yes], [14, 7, 6, 6, Yes], [14, 8, 5, 5, Yes], [14, 9, 4, 4, Yes], [14, 10, 3, 3, Yes], [14, 11, 2, 2, Yes], [14, 12, 1, 1, Yes], [14, 13, 0, 0, Yes], [14, 14, 26, 26, Yes], [14, 15, 25, 25, Yes], [14, 16, 24, 24, Yes], [14, 17, 23, 23, Yes], [14, 18, 22, 22, Yes], [14, 19, 21, 21, Yes], [14, 20, 20, 20, Yes], [14, 21, 19, 19, Yes], [14, 22, 18, 18, Yes], [14, 23, 17, 17, Yes], [14, 24, 16, 16, Yes], [14, 25, 15, 15, Yes], [14, 26, 14, 14, Yes], [15, 0, 12, 12, Yes], [15, 1, 11, 11, Yes], [15, 2, 10, 10, Yes], [15, 3, 9, 9, Yes], [15, 4, 8, 8, Yes], [15, 5, 7, 7, Yes], [15, 6, 6, 6, Yes], [15, 7, 5, 5, Yes], [15, 8, 4, 4, Yes], [15, 9, 3, 3, Yes], [15, 10, 2, 2, Yes], [15, 11, 1, 1, Yes], [15, 12, 0, 0, Yes], [15, 13, 26, 26, Yes], [15, 14, 25, 25, Yes], [15, 15, 24, 24, Yes], [15, 16, 23, 23, Yes], [15, 17, 22, 22, Yes], [15, 18, 21, 21, Yes], [15, 19, 20, 20, Yes], [15, 20, 19, 19, Yes], [15, 21, 18, 18, Yes], [15, 22, 17, 17, Yes], [15, 23, 16, 16, Yes], [15, 24, 15, 15, Yes], [15, 25, 14, 14, Yes], [15, 26, 13, 13, Yes], [16, 0, 11, 11, Yes], [16, 1, 10, 10, Yes], [16, 2, 9, 9, Yes], [16, 3, 8, 8, Yes], [16, 4, 7, 7, Yes], [16, 5, 6, 6, Yes], [16, 6, 5, 5, Yes], [16, 7, 4, 4, Yes], [16, 8, 3, 3, Yes], [16, 9, 2, 2, Yes], [16, 10, 1, 1, Yes], [16, 11, 0, 0, Yes], [16, 12, 26, 26, Yes], [16, 13, 25, 25, Yes], [16, 14, 24, 24, Yes], [16, 15, 23, 23, Yes], [16, 16, 22, 22, Yes], [16, 17, 21, 21, Yes], [16, 18, 20, 20, Yes], [16, 19, 19, 19, Yes], [16, 20, 18, 18, Yes], [16, 21, 17, 17, Yes], [16, 22, 16, 16, Yes], [16, 23, 15, 15, Yes], [16, 24, 14, 14, Yes], [16, 25, 13, 13, Yes], [16, 26, 12, 12, Yes], [17, 0, 10, 10, Yes], [17, 1, 9, 9, Yes], [17, 2, 8, 8, Yes], [17, 3, 7, 7, Yes], [17, 4, 6, 6, Yes], [17, 5, 5, 5, Yes], [17, 6, 4, 4, Yes], [17, 7, 3, 3, Yes], [17, 8, 2, 2, Yes], [17, 9, 1, 1, Yes], [17, 10, 0, 0, Yes], [17, 11, 26, 26, Yes], [17, 12, 25, 25, Yes], [17, 13, 24, 24, Yes], [17, 14, 23, 23, Yes], [17, 15, 22, 22, Yes], [17, 16, 21, 21, Yes], [17, 17, 20, 20, Yes], [17, 18, 19, 19, Yes], [17, 19, 18, 18, Yes], [17, 20, 17, 17, Yes], [17, 21, 16, 16, Yes], [17, 22, 15, 15, Yes], [17, 23, 14, 14, Yes], [17, 24, 13, 13, Yes], [17, 25, 12, 12, Yes], [17, 26, 11, 11, Yes], [18, 0, 9, 9, Yes], [18, 1, 8, 8, Yes], [18, 2, 7, 7, Yes], [18, 3, 6, 6, Yes], [18, 4, 5, 5, Yes], [18, 5, 4, 4, Yes], [18, 6, 3, 3, Yes], [18, 7, 2, 2, Yes], [18, 8, 1, 1, Yes], [18, 9, 0, 0, Yes], [18, 10, 26, 26, Yes], [18, 11, 25, 25, Yes], [18, 12, 24, 24, Yes], [18, 13, 23, 23, Yes], [18, 14, 22, 22, Yes], [18, 15, 21, 21, Yes], [18, 16, 20, 20, Yes], [18, 17, 19, 19, Yes], [18, 18, 18, 18, Yes], [18, 19, 17, 17, Yes], [18, 20, 16, 16, Yes], [18, 21, 15, 15, Yes], [18, 22, 14, 14, Yes], [18, 23, 13, 13, Yes], [18, 24, 12, 12, Yes], [18, 25, 11, 11, Yes], [18, 26, 10, 10, Yes], [19, 0, 8, 8, Yes], [19, 1, 7, 7, Yes], [19, 2, 6, 6, Yes], [19, 3, 5, 5, Yes], [19, 4, 4, 4, Yes], [19, 5, 3, 3, Yes], [19, 6, 2, 2, Yes], [19, 7, 1, 1, Yes], [19, 8, 0, 0, Yes], [19, 9, 26, 26, Yes], [19, 10, 25, 25, Yes], [19, 11, 24, 24, Yes], [19, 12, 23, 23, Yes], [19, 13, 22, 22, Yes], [19, 14, 21, 21, Yes], [19, 15, 20, 20, Yes], [19, 16, 19, 19, Yes], [19, 17, 18, 18, Yes], [19, 18, 17, 17, Yes], [19, 19, 16, 16, Yes], [19, 20, 15, 15, Yes], [19, 21, 14, 14, Yes], [19, 22, 13, 13, Yes], [19, 23, 12, 12, Yes], [19, 24, 11, 11, Yes], [19, 25, 10, 10, Yes], [19, 26, 9, 9, Yes], [20, 0, 7, 7, Yes], [20, 1, 6, 6, Yes], [20, 2, 5, 5, Yes], [20, 3, 4, 4, Yes], [20, 4, 3, 3, Yes], [20, 5, 2, 2, Yes], [20, 6, 1, 1, Yes], [20, 7, 0, 0, Yes], [20, 8, 26, 26, Yes], [20, 9, 25, 25, Yes], [20, 10, 24, 24, Yes], [20, 11, 23, 23, Yes], [20, 12, 22, 22, Yes], [20, 13, 21, 21, Yes], [20, 14, 20, 20, Yes], [20, 15, 19, 19, Yes], [20, 16, 18, 18, Yes], [20, 17, 17, 17, Yes], [20, 18, 16, 16, Yes], [20, 19, 15, 15, Yes], [20, 20, 14, 14, Yes], [20, 21, 13, 13, Yes], [20, 22, 12, 12, Yes], [20, 23, 11, 11, Yes], [20, 24, 10, 10, Yes], [20, 25, 9, 9, Yes], [20, 26, 8, 8, Yes], [21, 0, 6, 6, Yes], [21, 1, 5, 5, Yes], [21, 2, 4, 4, Yes], [21, 3, 3, 3, Yes], [21, 4, 2, 2, Yes], [21, 5, 1, 1, Yes], [21, 6, 0, 0, Yes], [21, 7, 26, 26, Yes], [21, 8, 25, 25, Yes], [21, 9, 24, 24, Yes], [21, 10, 23, 23, Yes], [21, 11, 22, 22, Yes], [21, 12, 21, 21, Yes], [21, 13, 20, 20, Yes], [21, 14, 19, 19, Yes], [21, 15, 18, 18, Yes], [21, 16, 17, 17, Yes], [21, 17, 16, 16, Yes], [21, 18, 15, 15, Yes], [21, 19, 14, 14, Yes], [21, 20, 13, 13, Yes], [21, 21, 12, 12, Yes], [21, 22, 11, 11, Yes], [21, 23, 10, 10, Yes], [21, 24, 9, 9, Yes], [21, 25, 8, 8, Yes], [21, 26, 7, 7, Yes], [22, 0, 5, 5, Yes], [22, 1, 4, 4, Yes], [22, 2, 3, 3, Yes], [22, 3, 2, 2, Yes], [22, 4, 1, 1, Yes], [22, 5, 0, 0, Yes], [22, 6, 26, 26, Yes], [22, 7, 25, 25, Yes], [22, 8, 24, 24, Yes], [22, 9, 23, 23, Yes], [22, 10, 22, 22, Yes], [22, 11, 21, 21, Yes], [22, 12, 20, 20, Yes], [22, 13, 19, 19, Yes], [22, 14, 18, 18, Yes], [22, 15, 17, 17, Yes], [22, 16, 16, 16, Yes], [22, 17, 15, 15, Yes], [22, 18, 14, 14, Yes], [22, 19, 13, 13, Yes], [22, 20, 12, 12, Yes], [22, 21, 11, 11, Yes], [22, 22, 10, 10, Yes], [22, 23, 9, 9, Yes], [22, 24, 8, 8, Yes], [22, 25, 7, 7, Yes], [22, 26, 6, 6, Yes], [23, 0, 4, 4, Yes], [23, 1, 3, 3, Yes], [23, 2, 2, 2, Yes], [23, 3, 1, 1, Yes], [23, 4, 0, 0, Yes], [23, 5, 26, 26, Yes], [23, 6, 25, 25, Yes], [23, 7, 24, 24, Yes], [23, 8, 23, 23, Yes], [23, 9, 22, 22, Yes], [23, 10, 21, 21, Yes], [23, 11, 20, 20, Yes], [23, 12, 19, 19, Yes], [23, 13, 18, 18, Yes], [23, 14, 17, 17, Yes], [23, 15, 16, 16, Yes], [23, 16, 15, 15, Yes], [23, 17, 14, 14, Yes], [23, 18, 13, 13, Yes], [23, 19, 12, 12, Yes], [23, 20, 11, 11, Yes], [23, 21, 10, 10, Yes], [23, 22, 9, 9, Yes], [23, 23, 8, 8, Yes], [23, 24, 7, 7, Yes], [23, 25, 6, 6, Yes], [23, 26, 5, 5, Yes], [24, 0, 3, 3, Yes], [24, 1, 2, 2, Yes], [24, 2, 1, 1, Yes], [24, 3, 0, 0, Yes], [24, 4, 26, 26, Yes], [24, 5, 25, 25, Yes], [24, 6, 24, 24, Yes], [24, 7, 23, 23, Yes], [24, 8, 22, 22, Yes], [24, 9, 21, 21, Yes], [24, 10, 20, 20, Yes], [24, 11, 19, 19, Yes], [24, 12, 18, 18, Yes], [24, 13, 17, 17, Yes], [24, 14, 16, 16, Yes], [24, 15, 15, 15, Yes], [24, 16, 14, 14, Yes], [24, 17, 13, 13, Yes], [24, 18, 12, 12, Yes], [24, 19, 11, 11, Yes], [24, 20, 10, 10, Yes], [24, 21, 9, 9, Yes], [24, 22, 8, 8, Yes], [24, 23, 7, 7, Yes], [24, 24, 6, 6, Yes], [24, 25, 5, 5, Yes], [24, 26, 4, 4, Yes], [25, 0, 2, 2, Yes], [25, 1, 1, 1, Yes], [25, 2, 0, 0, Yes], [25, 3, 26, 26, Yes], [25, 4, 25, 25, Yes], [25, 5, 24, 24, Yes], [25, 6, 23, 23, Yes], [25, 7, 22, 22, Yes], [25, 8, 21, 21, Yes], [25, 9, 20, 20, Yes], [25, 10, 19, 19, Yes], [25, 11, 18, 18, Yes], [25, 12, 17, 17, Yes], [25, 13, 16, 16, Yes], [25, 14, 15, 15, Yes], [25, 15, 14, 14, Yes], [25, 16, 13, 13, Yes], [25, 17, 12, 12, Yes], [25, 18, 11, 11, Yes], [25, 19, 10, 10, Yes], [25, 20, 9, 9, Yes], [25, 21, 8, 8, Yes], [25, 22, 7, 7, Yes], [25, 23, 6, 6, Yes], [25, 24, 5, 5, Yes], [25, 25, 4, 4, Yes], [25, 26, 3, 3, Yes], [26, 0, 1, 1, Yes], [26, 1, 0, 0, Yes], [26, 2, 26, 26, Yes], [26, 3, 25, 25, Yes], [26, 4, 24, 24, Yes], [26, 5, 23, 23, Yes], [26, 6, 22, 22, Yes], [26, 7, 21, 21, Yes], [26, 8, 20, 20, Yes], [26, 9, 19, 19, Yes], [26, 10, 18, 18, Yes], [26, 11, 17, 17, Yes], [26, 12, 16, 16, Yes], [26, 13, 15, 15, Yes], [26, 14, 14, 14, Yes], [26, 15, 13, 13, Yes], [26, 16, 12, 12, Yes], [26, 17, 11, 11, Yes], [26, 18, 10, 10, Yes], [26, 19, 9, 9, Yes], [26, 20, 8, 8, Yes], [26, 21, 7, 7, Yes], [26, 22, 6, 6, Yes], [26, 23, 5, 5, Yes], [26, 24, 4, 4, Yes], [26, 25, 3, 3, Yes], [26, 26, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)","explanation":"List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (27, 27, [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 26, 26, Yes], [0, 2, 25, 25, Yes], [0, 3, 24, 24, Yes], [0, 4, 23, 23, Yes], [0, 5, 22, 22, Yes], [0, 6, 21, 21, Yes], [0, 7, 20, 20, Yes], [0, 8, 19, 19, Yes], [0, 9, 18, 18, Yes], [0, 10, 17, 17, Yes], [0, 11, 16, 16, Yes], [0, 12, 15, 15, Yes], [0, 13, 14, 14, Yes], [0, 14, 13, 13, Yes], [0, 15, 12, 12, Yes], [0, 16, 11, 11, Yes], [0, 17, 10, 10, Yes], [0, 18, 9, 9, Yes], [0, 19, 8, 8, Yes], [0, 20, 7, 7, Yes], [0, 21, 6, 6, Yes], [0, 22, 5, 5, Yes], [0, 23, 4, 4, Yes], [0, 24, 3, 3, Yes], [0, 25, 2, 2, Yes], [0, 26, 1, 1, Yes], [1, 0, 26, 26, Yes], [1, 1, 25, 25, Yes], [1, 2, 24, 24, Yes], [1, 3, 23, 23, Yes], [1, 4, 22, 22, Yes], [1, 5, 21, 21, Yes], [1, 6, 20, 20, Yes], [1, 7, 19, 19, Yes], [1, 8, 18, 18, Yes], [1, 9, 17, 17, Yes], [1, 10, 16, 16, Yes], [1, 11, 15, 15, Yes], [1, 12, 14, 14, Yes], [1, 13, 13, 13, Yes], [1, 14, 12, 12, Yes], [1, 15, 11, 11, Yes], [1, 16, 10, 10, Yes], [1, 17, 9, 9, Yes], [1, 18, 8, 8, Yes], [1, 19, 7, 7, Yes], [1, 20, 6, 6, Yes], [1, 21, 5, 5, Yes], [1, 22, 4, 4, Yes], [1, 23, 3, 3, Yes], [1, 24, 2, 2, Yes], [1, 25, 1, 1, Yes], [1, 26, 0, 0, Yes], [2, 0, 25, 25, Yes], [2, 1, 24, 24, Yes], [2, 2, 23, 23, Yes], [2, 3, 22, 22, Yes], [2, 4, 21, 21, Yes], [2, 5, 20, 20, Yes], [2, 6, 19, 19, Yes], [2, 7, 18, 18, Yes], [2, 8, 17, 17, Yes], [2, 9, 16, 16, Yes], [2, 10, 15, 15, Yes], [2, 11, 14, 14, Yes], [2, 12, 13, 13, Yes], [2, 13, 12, 12, Yes], [2, 14, 11, 11, Yes], [2, 15, 10, 10, Yes], [2, 16, 9, 9, Yes], [2, 17, 8, 8, Yes], [2, 18, 7, 7, Yes], [2, 19, 6, 6, Yes], [2, 20, 5, 5, Yes], [2, 21, 4, 4, Yes], [2, 22, 3, 3, Yes], [2, 23, 2, 2, Yes], [2, 24, 1, 1, Yes], [2, 25, 0, 0, Yes], [2, 26, 26, 26, Yes], [3, 0, 24, 24, Yes], [3, 1, 23, 23, Yes], [3, 2, 22, 22, Yes], [3, 3, 21, 21, Yes], [3, 4, 20, 20, Yes], [3, 5, 19, 19, Yes], [3, 6, 18, 18, Yes], [3, 7, 17, 17, Yes], [3, 8, 16, 16, Yes], [3, 9, 15, 15, Yes], [3, 10, 14, 14, Yes], [3, 11, 13, 13, Yes], [3, 12, 12, 12, Yes], [3, 13, 11, 11, Yes], [3, 14, 10, 10, Yes], [3, 15, 9, 9, Yes], [3, 16, 8, 8, Yes], [3, 17, 7, 7, Yes], [3, 18, 6, 6, Yes], [3, 19, 5, 5, Yes], [3, 20, 4, 4, Yes], [3, 21, 3, 3, Yes], [3, 22, 2, 2, Yes], [3, 23, 1, 1, Yes], [3, 24, 0, 0, Yes], [3, 25, 26, 26, Yes], [3, 26, 25, 25, Yes], [4, 0, 23, 23, Yes], [4, 1, 22, 22, Yes], [4, 2, 21, 21, Yes], [4, 3, 20, 20, Yes], [4, 4, 19, 19, Yes], [4, 5, 18, 18, Yes], [4, 6, 17, 17, Yes], [4, 7, 16, 16, Yes], [4, 8, 15, 15, Yes], [4, 9, 14, 14, Yes], [4, 10, 13, 13, Yes], [4, 11, 12, 12, Yes], [4, 12, 11, 11, Yes], [4, 13, 10, 10, Yes], [4, 14, 9, 9, Yes], [4, 15, 8, 8, Yes], [4, 16, 7, 7, Yes], [4, 17, 6, 6, Yes], [4, 18, 5, 5, Yes], [4, 19, 4, 4, Yes], [4, 20, 3, 3, Yes], [4, 21, 2, 2, Yes], [4, 22, 1, 1, Yes], [4, 23, 0, 0, Yes], [4, 24, 26, 26, Yes], [4, 25, 25, 25, Yes], [4, 26, 24, 24, Yes], [5, 0, 22, 22, Yes], [5, 1, 21, 21, Yes], [5, 2, 20, 20, Yes], [5, 3, 19, 19, Yes], [5, 4, 18, 18, Yes], [5, 5, 17, 17, Yes], [5, 6, 16, 16, Yes], [5, 7, 15, 15, Yes], [5, 8, 14, 14, Yes], [5, 9, 13, 13, Yes], [5, 10, 12, 12, Yes], [5, 11, 11, 11, Yes], [5, 12, 10, 10, Yes], [5, 13, 9, 9, Yes], [5, 14, 8, 8, Yes], [5, 15, 7, 7, Yes], [5, 16, 6, 6, Yes], [5, 17, 5, 5, Yes], [5, 18, 4, 4, Yes], [5, 19, 3, 3, Yes], [5, 20, 2, 2, Yes], [5, 21, 1, 1, Yes], [5, 22, 0, 0, Yes], [5, 23, 26, 26, Yes], [5, 24, 25, 25, Yes], [5, 25, 24, 24, Yes], [5, 26, 23, 23, Yes], [6, 0, 21, 21, Yes], [6, 1, 20, 20, Yes], [6, 2, 19, 19, Yes], [6, 3, 18, 18, Yes], [6, 4, 17, 17, Yes], [6, 5, 16, 16, Yes], [6, 6, 15, 15, Yes], [6, 7, 14, 14, Yes], [6, 8, 13, 13, Yes], [6, 9, 12, 12, Yes], [6, 10, 11, 11, Yes], [6, 11, 10, 10, Yes], [6, 12, 9, 9, Yes], [6, 13, 8, 8, Yes], [6, 14, 7, 7, Yes], [6, 15, 6, 6, Yes], [6, 16, 5, 5, Yes], [6, 17, 4, 4, Yes], [6, 18, 3, 3, Yes], [6, 19, 2, 2, Yes], [6, 20, 1, 1, Yes], [6, 21, 0, 0, Yes], [6, 22, 26, 26, Yes], [6, 23, 25, 25, Yes], [6, 24, 24, 24, Yes], [6, 25, 23, 23, Yes], [6, 26, 22, 22, Yes], [7, 0, 20, 20, Yes], [7, 1, 19, 19, Yes], [7, 2, 18, 18, Yes], [7, 3, 17, 17, Yes], [7, 4, 16, 16, Yes], [7, 5, 15, 15, Yes], [7, 6, 14, 14, Yes], [7, 7, 13, 13, Yes], [7, 8, 12, 12, Yes], [7, 9, 11, 11, Yes], [7, 10, 10, 10, Yes], [7, 11, 9, 9, Yes], [7, 12, 8, 8, Yes], [7, 13, 7, 7, Yes], [7, 14, 6, 6, Yes], [7, 15, 5, 5, Yes], [7, 16, 4, 4, Yes], [7, 17, 3, 3, Yes], [7, 18, 2, 2, Yes], [7, 19, 1, 1, Yes], [7, 20, 0, 0, Yes], [7, 21, 26, 26, Yes], [7, 22, 25, 25, Yes], [7, 23, 24, 24, Yes], [7, 24, 23, 23, Yes], [7, 25, 22, 22, Yes], [7, 26, 21, 21, Yes], [8, 0, 19, 19, Yes], [8, 1, 18, 18, Yes], [8, 2, 17, 17, Yes], [8, 3, 16, 16, Yes], [8, 4, 15, 15, Yes], [8, 5, 14, 14, Yes], [8, 6, 13, 13, Yes], [8, 7, 12, 12, Yes], [8, 8, 11, 11, Yes], [8, 9, 10, 10, Yes], [8, 10, 9, 9, Yes], [8, 11, 8, 8, Yes], [8, 12, 7, 7, Yes], [8, 13, 6, 6, Yes], [8, 14, 5, 5, Yes], [8, 15, 4, 4, Yes], [8, 16, 3, 3, Yes], [8, 17, 2, 2, Yes], [8, 18, 1, 1, Yes], [8, 19, 0, 0, Yes], [8, 20, 26, 26, Yes], [8, 21, 25, 25, Yes], [8, 22, 24, 24, Yes], [8, 23, 23, 23, Yes], [8, 24, 22, 22, Yes], [8, 25, 21, 21, Yes], [8, 26, 20, 20, Yes], [9, 0, 18, 18, Yes], [9, 1, 17, 17, Yes], [9, 2, 16, 16, Yes], [9, 3, 15, 15, Yes], [9, 4, 14, 14, Yes], [9, 5, 13, 13, Yes], [9, 6, 12, 12, Yes], [9, 7, 11, 11, Yes], [9, 8, 10, 10, Yes], [9, 9, 9, 9, Yes], [9, 10, 8, 8, Yes], [9, 11, 7, 7, Yes], [9, 12, 6, 6, Yes], [9, 13, 5, 5, Yes], [9, 14, 4, 4, Yes], [9, 15, 3, 3, Yes], [9, 16, 2, 2, Yes], [9, 17, 1, 1, Yes], [9, 18, 0, 0, Yes], [9, 19, 26, 26, Yes], [9, 20, 25, 25, Yes], [9, 21, 24, 24, Yes], [9, 22, 23, 23, Yes], [9, 23, 22, 22, Yes], [9, 24, 21, 21, Yes], [9, 25, 20, 20, Yes], [9, 26, 19, 19, Yes], [10, 0, 17, 17, Yes], [10, 1, 16, 16, Yes], [10, 2, 15, 15, Yes], [10, 3, 14, 14, Yes], [10, 4, 13, 13, Yes], [10, 5, 12, 12, Yes], [10, 6, 11, 11, Yes], [10, 7, 10, 10, Yes], [10, 8, 9, 9, Yes], [10, 9, 8, 8, Yes], [10, 10, 7, 7, Yes], [10, 11, 6, 6, Yes], [10, 12, 5, 5, Yes], [10, 13, 4, 4, Yes], [10, 14, 3, 3, Yes], [10, 15, 2, 2, Yes], [10, 16, 1, 1, Yes], [10, 17, 0, 0, Yes], [10, 18, 26, 26, Yes], [10, 19, 25, 25, Yes], [10, 20, 24, 24, Yes], [10, 21, 23, 23, Yes], [10, 22, 22, 22, Yes], [10, 23, 21, 21, Yes], [10, 24, 20, 20, Yes], [10, 25, 19, 19, Yes], [10, 26, 18, 18, Yes], [11, 0, 16, 16, Yes], [11, 1, 15, 15, Yes], [11, 2, 14, 14, Yes], [11, 3, 13, 13, Yes], [11, 4, 12, 12, Yes], [11, 5, 11, 11, Yes], [11, 6, 10, 10, Yes], [11, 7, 9, 9, Yes], [11, 8, 8, 8, Yes], [11, 9, 7, 7, Yes], [11, 10, 6, 6, Yes], [11, 11, 5, 5, Yes], [11, 12, 4, 4, Yes], [11, 13, 3, 3, Yes], [11, 14, 2, 2, Yes], [11, 15, 1, 1, Yes], [11, 16, 0, 0, Yes], [11, 17, 26, 26, Yes], [11, 18, 25, 25, Yes], [11, 19, 24, 24, Yes], [11, 20, 23, 23, Yes], [11, 21, 22, 22, Yes], [11, 22, 21, 21, Yes], [11, 23, 20, 20, Yes], [11, 24, 19, 19, Yes], [11, 25, 18, 18, Yes], [11, 26, 17, 17, Yes], [12, 0, 15, 15, Yes], [12, 1, 14, 14, Yes], [12, 2, 13, 13, Yes], [12, 3, 12, 12, Yes], [12, 4, 11, 11, Yes], [12, 5, 10, 10, Yes], [12, 6, 9, 9, Yes], [12, 7, 8, 8, Yes], [12, 8, 7, 7, Yes], [12, 9, 6, 6, Yes], [12, 10, 5, 5, Yes], [12, 11, 4, 4, Yes], [12, 12, 3, 3, Yes], [12, 13, 2, 2, Yes], [12, 14, 1, 1, Yes], [12, 15, 0, 0, Yes], [12, 16, 26, 26, Yes], [12, 17, 25, 25, Yes], [12, 18, 24, 24, Yes], [12, 19, 23, 23, Yes], [12, 20, 22, 22, Yes], [12, 21, 21, 21, Yes], [12, 22, 20, 20, Yes], [12, 23, 19, 19, Yes], [12, 24, 18, 18, Yes], [12, 25, 17, 17, Yes], [12, 26, 16, 16, Yes], [13, 0, 14, 14, Yes], [13, 1, 13, 13, Yes], [13, 2, 12, 12, Yes], [13, 3, 11, 11, Yes], [13, 4, 10, 10, Yes], [13, 5, 9, 9, Yes], [13, 6, 8, 8, Yes], [13, 7, 7, 7, Yes], [13, 8, 6, 6, Yes], [13, 9, 5, 5, Yes], [13, 10, 4, 4, Yes], [13, 11, 3, 3, Yes], [13, 12, 2, 2, Yes], [13, 13, 1, 1, Yes], [13, 14, 0, 0, Yes], [13, 15, 26, 26, Yes], [13, 16, 25, 25, Yes], [13, 17, 24, 24, Yes], [13, 18, 23, 23, Yes], [13, 19, 22, 22, Yes], [13, 20, 21, 21, Yes], [13, 21, 20, 20, Yes], [13, 22, 19, 19, Yes], [13, 23, 18, 18, Yes], [13, 24, 17, 17, Yes], [13, 25, 16, 16, Yes], [13, 26, 15, 15, Yes], [14, 0, 13, 13, Yes], [14, 1, 12, 12, Yes], [14, 2, 11, 11, Yes], [14, 3, 10, 10, Yes], [14, 4, 9, 9, Yes], [14, 5, 8, 8, Yes], [14, 6, 7, 7, Yes], [14, 7, 6, 6, Yes], [14, 8, 5, 5, Yes], [14, 9, 4, 4, Yes], [14, 10, 3, 3, Yes], [14, 11, 2, 2, Yes], [14, 12, 1, 1, Yes], [14, 13, 0, 0, Yes], [14, 14, 26, 26, Yes], [14, 15, 25, 25, Yes], [14, 16, 24, 24, Yes], [14, 17, 23, 23, Yes], [14, 18, 22, 22, Yes], [14, 19, 21, 21, Yes], [14, 20, 20, 20, Yes], [14, 21, 19, 19, Yes], [14, 22, 18, 18, Yes], [14, 23, 17, 17, Yes], [14, 24, 16, 16, Yes], [14, 25, 15, 15, Yes], [14, 26, 14, 14, Yes], [15, 0, 12, 12, Yes], [15, 1, 11, 11, Yes], [15, 2, 10, 10, Yes], [15, 3, 9, 9, Yes], [15, 4, 8, 8, Yes], [15, 5, 7, 7, Yes], [15, 6, 6, 6, Yes], [15, 7, 5, 5, Yes], [15, 8, 4, 4, Yes], [15, 9, 3, 3, Yes], [15, 10, 2, 2, Yes], [15, 11, 1, 1, Yes], [15, 12, 0, 0, Yes], [15, 13, 26, 26, Yes], [15, 14, 25, 25, Yes], [15, 15, 24, 24, Yes], [15, 16, 23, 23, Yes], [15, 17, 22, 22, Yes], [15, 18, 21, 21, Yes], [15, 19, 20, 20, Yes], [15, 20, 19, 19, Yes], [15, 21, 18, 18, Yes], [15, 22, 17, 17, Yes], [15, 23, 16, 16, Yes], [15, 24, 15, 15, Yes], [15, 25, 14, 14, Yes], [15, 26, 13, 13, Yes], [16, 0, 11, 11, Yes], [16, 1, 10, 10, Yes], [16, 2, 9, 9, Yes], [16, 3, 8, 8, Yes], [16, 4, 7, 7, Yes], [16, 5, 6, 6, Yes], [16, 6, 5, 5, Yes], [16, 7, 4, 4, Yes], [16, 8, 3, 3, Yes], [16, 9, 2, 2, Yes], [16, 10, 1, 1, Yes], [16, 11, 0, 0, Yes], [16, 12, 26, 26, Yes], [16, 13, 25, 25, Yes], [16, 14, 24, 24, Yes], [16, 15, 23, 23, Yes], [16, 16, 22, 22, Yes], [16, 17, 21, 21, Yes], [16, 18, 20, 20, Yes], [16, 19, 19, 19, Yes], [16, 20, 18, 18, Yes], [16, 21, 17, 17, Yes], [16, 22, 16, 16, Yes], [16, 23, 15, 15, Yes], [16, 24, 14, 14, Yes], [16, 25, 13, 13, Yes], [16, 26, 12, 12, Yes], [17, 0, 10, 10, Yes], [17, 1, 9, 9, Yes], [17, 2, 8, 8, Yes], [17, 3, 7, 7, Yes], [17, 4, 6, 6, Yes], [17, 5, 5, 5, Yes], [17, 6, 4, 4, Yes], [17, 7, 3, 3, Yes], [17, 8, 2, 2, Yes], [17, 9, 1, 1, Yes], [17, 10, 0, 0, Yes], [17, 11, 26, 26, Yes], [17, 12, 25, 25, Yes], [17, 13, 24, 24, Yes], [17, 14, 23, 23, Yes], [17, 15, 22, 22, Yes], [17, 16, 21, 21, Yes], [17, 17, 20, 20, Yes], [17, 18, 19, 19, Yes], [17, 19, 18, 18, Yes], [17, 20, 17, 17, Yes], [17, 21, 16, 16, Yes], [17, 22, 15, 15, Yes], [17, 23, 14, 14, Yes], [17, 24, 13, 13, Yes], [17, 25, 12, 12, Yes], [17, 26, 11, 11, Yes], [18, 0, 9, 9, Yes], [18, 1, 8, 8, Yes], [18, 2, 7, 7, Yes], [18, 3, 6, 6, Yes], [18, 4, 5, 5, Yes], [18, 5, 4, 4, Yes], [18, 6, 3, 3, Yes], [18, 7, 2, 2, Yes], [18, 8, 1, 1, Yes], [18, 9, 0, 0, Yes], [18, 10, 26, 26, Yes], [18, 11, 25, 25, Yes], [18, 12, 24, 24, Yes], [18, 13, 23, 23, Yes], [18, 14, 22, 22, Yes], [18, 15, 21, 21, Yes], [18, 16, 20, 20, Yes], [18, 17, 19, 19, Yes], [18, 18, 18, 18, Yes], [18, 19, 17, 17, Yes], [18, 20, 16, 16, Yes], [18, 21, 15, 15, Yes], [18, 22, 14, 14, Yes], [18, 23, 13, 13, Yes], [18, 24, 12, 12, Yes], [18, 25, 11, 11, Yes], [18, 26, 10, 10, Yes], [19, 0, 8, 8, Yes], [19, 1, 7, 7, Yes], [19, 2, 6, 6, Yes], [19, 3, 5, 5, Yes], [19, 4, 4, 4, Yes], [19, 5, 3, 3, Yes], [19, 6, 2, 2, Yes], [19, 7, 1, 1, Yes], [19, 8, 0, 0, Yes], [19, 9, 26, 26, Yes], [19, 10, 25, 25, Yes], [19, 11, 24, 24, Yes], [19, 12, 23, 23, Yes], [19, 13, 22, 22, Yes], [19, 14, 21, 21, Yes], [19, 15, 20, 20, Yes], [19, 16, 19, 19, Yes], [19, 17, 18, 18, Yes], [19, 18, 17, 17, Yes], [19, 19, 16, 16, Yes], [19, 20, 15, 15, Yes], [19, 21, 14, 14, Yes], [19, 22, 13, 13, Yes], [19, 23, 12, 12, Yes], [19, 24, 11, 11, Yes], [19, 25, 10, 10, Yes], [19, 26, 9, 9, Yes], [20, 0, 7, 7, Yes], [20, 1, 6, 6, Yes], [20, 2, 5, 5, Yes], [20, 3, 4, 4, Yes], [20, 4, 3, 3, Yes], [20, 5, 2, 2, Yes], [20, 6, 1, 1, Yes], [20, 7, 0, 0, Yes], [20, 8, 26, 26, Yes], [20, 9, 25, 25, Yes], [20, 10, 24, 24, Yes], [20, 11, 23, 23, Yes], [20, 12, 22, 22, Yes], [20, 13, 21, 21, Yes], [20, 14, 20, 20, Yes], [20, 15, 19, 19, Yes], [20, 16, 18, 18, Yes], [20, 17, 17, 17, Yes], [20, 18, 16, 16, Yes], [20, 19, 15, 15, Yes], [20, 20, 14, 14, Yes], [20, 21, 13, 13, Yes], [20, 22, 12, 12, Yes], [20, 23, 11, 11, Yes], [20, 24, 10, 10, Yes], [20, 25, 9, 9, Yes], [20, 26, 8, 8, Yes], [21, 0, 6, 6, Yes], [21, 1, 5, 5, Yes], [21, 2, 4, 4, Yes], [21, 3, 3, 3, Yes], [21, 4, 2, 2, Yes], [21, 5, 1, 1, Yes], [21, 6, 0, 0, Yes], [21, 7, 26, 26, Yes], [21, 8, 25, 25, Yes], [21, 9, 24, 24, Yes], [21, 10, 23, 23, Yes], [21, 11, 22, 22, Yes], [21, 12, 21, 21, Yes], [21, 13, 20, 20, Yes], [21, 14, 19, 19, Yes], [21, 15, 18, 18, Yes], [21, 16, 17, 17, Yes], [21, 17, 16, 16, Yes], [21, 18, 15, 15, Yes], [21, 19, 14, 14, Yes], [21, 20, 13, 13, Yes], [21, 21, 12, 12, Yes], [21, 22, 11, 11, Yes], [21, 23, 10, 10, Yes], [21, 24, 9, 9, Yes], [21, 25, 8, 8, Yes], [21, 26, 7, 7, Yes], [22, 0, 5, 5, Yes], [22, 1, 4, 4, Yes], [22, 2, 3, 3, Yes], [22, 3, 2, 2, Yes], [22, 4, 1, 1, Yes], [22, 5, 0, 0, Yes], [22, 6, 26, 26, Yes], [22, 7, 25, 25, Yes], [22, 8, 24, 24, Yes], [22, 9, 23, 23, Yes], [22, 10, 22, 22, Yes], [22, 11, 21, 21, Yes], [22, 12, 20, 20, Yes], [22, 13, 19, 19, Yes], [22, 14, 18, 18, Yes], [22, 15, 17, 17, Yes], [22, 16, 16, 16, Yes], [22, 17, 15, 15, Yes], [22, 18, 14, 14, Yes], [22, 19, 13, 13, Yes], [22, 20, 12, 12, Yes], [22, 21, 11, 11, Yes], [22, 22, 10, 10, Yes], [22, 23, 9, 9, Yes], [22, 24, 8, 8, Yes], [22, 25, 7, 7, Yes], [22, 26, 6, 6, Yes], [23, 0, 4, 4, Yes], [23, 1, 3, 3, Yes], [23, 2, 2, 2, Yes], [23, 3, 1, 1, Yes], [23, 4, 0, 0, Yes], [23, 5, 26, 26, Yes], [23, 6, 25, 25, Yes], [23, 7, 24, 24, Yes], [23, 8, 23, 23, Yes], [23, 9, 22, 22, Yes], [23, 10, 21, 21, Yes], [23, 11, 20, 20, Yes], [23, 12, 19, 19, Yes], [23, 13, 18, 18, Yes], [23, 14, 17, 17, Yes], [23, 15, 16, 16, Yes], [23, 16, 15, 15, Yes], [23, 17, 14, 14, Yes], [23, 18, 13, 13, Yes], [23, 19, 12, 12, Yes], [23, 20, 11, 11, Yes], [23, 21, 10, 10, Yes], [23, 22, 9, 9, Yes], [23, 23, 8, 8, Yes], [23, 24, 7, 7, Yes], [23, 25, 6, 6, Yes], [23, 26, 5, 5, Yes], [24, 0, 3, 3, Yes], [24, 1, 2, 2, Yes], [24, 2, 1, 1, Yes], [24, 3, 0, 0, Yes], [24, 4, 26, 26, Yes], [24, 5, 25, 25, Yes], [24, 6, 24, 24, Yes], [24, 7, 23, 23, Yes], [24, 8, 22, 22, Yes], [24, 9, 21, 21, Yes], [24, 10, 20, 20, Yes], [24, 11, 19, 19, Yes], [24, 12, 18, 18, Yes], [24, 13, 17, 17, Yes], [24, 14, 16, 16, Yes], [24, 15, 15, 15, Yes], [24, 16, 14, 14, Yes], [24, 17, 13, 13, Yes], [24, 18, 12, 12, Yes], [24, 19, 11, 11, Yes], [24, 20, 10, 10, Yes], [24, 21, 9, 9, Yes], [24, 22, 8, 8, Yes], [24, 23, 7, 7, Yes], [24, 24, 6, 6, Yes], [24, 25, 5, 5, Yes], [24, 26, 4, 4, Yes], [25, 0, 2, 2, Yes], [25, 1, 1, 1, Yes], [25, 2, 0, 0, Yes], [25, 3, 26, 26, Yes], [25, 4, 25, 25, Yes], [25, 5, 24, 24, Yes], [25, 6, 23, 23, Yes], [25, 7, 22, 22, Yes], [25, 8, 21, 21, Yes], [25, 9, 20, 20, Yes], [25, 10, 19, 19, Yes], [25, 11, 18, 18, Yes], [25, 12, 17, 17, Yes], [25, 13, 16, 16, Yes], [25, 14, 15, 15, Yes], [25, 15, 14, 14, Yes], [25, 16, 13, 13, Yes], [25, 17, 12, 12, Yes], [25, 18, 11, 11, Yes], [25, 19, 10, 10, Yes], [25, 20, 9, 9, Yes], [25, 21, 8, 8, Yes], [25, 22, 7, 7, Yes], [25, 23, 6, 6, Yes], [25, 24, 5, 5, Yes], [25, 25, 4, 4, Yes], [25, 26, 3, 3, Yes], [26, 0, 1, 1, Yes], [26, 1, 0, 0, Yes], [26, 2, 26, 26, Yes], [26, 3, 25, 25, Yes], [26, 4, 24, 24, Yes], [26, 5, 23, 23, Yes], [26, 6, 22, 22, Yes], [26, 7, 21, 21, Yes], [26, 8, 20, 20, Yes], [26, 9, 19, 19, Yes], [26, 10, 18, 18, Yes], [26, 11, 17, 17, Yes], [26, 12, 16, 16, Yes], [26, 13, 15, 15, Yes], [26, 14, 14, 14, Yes], [26, 15, 13, 13, Yes], [26, 16, 12, 12, Yes], [26, 17, 11, 11, Yes], [26, 18, 10, 10, Yes], [26, 19, 9, 9, Yes], [26, 20, 8, 8, Yes], [26, 21, 7, 7, Yes], [26, 22, 6, 6, Yes], [26, 23, 5, 5, Yes], [26, 24, 4, 4, Yes], [26, 25, 3, 3, Yes], [26, 26, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)."},"skills":["Qualify Both Groups and Every Product Pair before Applying Kernel Theorems"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0026","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"742560a493b20086190d92516d0d898ca84b19e63450c065b782d0d2507a2f49","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141026,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2], Yes, Yes, Yes, [[0, [0, 2], Yes], [1, [0, 2], Yes], [2, [0, 2], Yes], [3, [0, 2], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2], Yes, Yes, Yes, [[0, [0, 2], Yes], [1, [0, 2], Yes], [2, [0, 2], Yes], [3, [0, 2], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0027","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"accec0c5ccc2670aeb86c094c8f053154e1165fed621b790ab40af55674a2cfd","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141027,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4], Yes, Yes, Yes, [[0, [0, 2, 4], Yes], [1, [0, 2, 4], Yes], [2, [0, 2, 4], Yes], [3, [0, 2, 4], Yes], [4, [0, 2, 4], Yes], [5, [0, 2, 4], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4], Yes, Yes, Yes, [[0, [0, 2, 4], Yes], [1, [0, 2, 4], Yes], [2, [0, 2, 4], Yes], [3, [0, 2, 4], Yes], [4, [0, 2, 4], Yes], [5, [0, 2, 4], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0028","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06b4a779eddad689940a252c5c41514367b766e478c6648f3ed07dae4eccddcf","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141028,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6], Yes, Yes, Yes, [[0, [0, 2, 4, 6], Yes], [1, [0, 2, 4, 6], Yes], [2, [0, 2, 4, 6], Yes], [3, [0, 2, 4, 6], Yes], [4, [0, 2, 4, 6], Yes], [5, [0, 2, 4, 6], Yes], [6, [0, 2, 4, 6], Yes], [7, [0, 2, 4, 6], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6], Yes, Yes, Yes, [[0, [0, 2, 4, 6], Yes], [1, [0, 2, 4, 6], Yes], [2, [0, 2, 4, 6], Yes], [3, [0, 2, 4, 6], Yes], [4, [0, 2, 4, 6], Yes], [5, [0, 2, 4, 6], Yes], [6, [0, 2, 4, 6], Yes], [7, [0, 2, 4, 6], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0029","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db77a74e5a45928d0575a2a09547fbdad7a06ebd86c2e352ca17f066e56d6f30","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141029,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8], Yes], [1, [0, 2, 4, 6, 8], Yes], [2, [0, 2, 4, 6, 8], Yes], [3, [0, 2, 4, 6, 8], Yes], [4, [0, 2, 4, 6, 8], Yes], [5, [0, 2, 4, 6, 8], Yes], [6, [0, 2, 4, 6, 8], Yes], [7, [0, 2, 4, 6, 8], Yes], [8, [0, 2, 4, 6, 8], Yes], [9, [0, 2, 4, 6, 8], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8], Yes], [1, [0, 2, 4, 6, 8], Yes], [2, [0, 2, 4, 6, 8], Yes], [3, [0, 2, 4, 6, 8], Yes], [4, [0, 2, 4, 6, 8], Yes], [5, [0, 2, 4, 6, 8], Yes], [6, [0, 2, 4, 6, 8], Yes], [7, [0, 2, 4, 6, 8], Yes], [8, [0, 2, 4, 6, 8], Yes], [9, [0, 2, 4, 6, 8], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0030","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02307eda6cab5d8677d3e445c06bb0e26edc6cbb72838e23bd894366272e4521","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141030,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10], Yes], [1, [0, 2, 4, 6, 8, 10], Yes], [2, [0, 2, 4, 6, 8, 10], Yes], [3, [0, 2, 4, 6, 8, 10], Yes], [4, [0, 2, 4, 6, 8, 10], Yes], [5, [0, 2, 4, 6, 8, 10], Yes], [6, [0, 2, 4, 6, 8, 10], Yes], [7, [0, 2, 4, 6, 8, 10], Yes], [8, [0, 2, 4, 6, 8, 10], Yes], [9, [0, 2, 4, 6, 8, 10], Yes], [10, [0, 2, 4, 6, 8, 10], Yes], [11, [0, 2, 4, 6, 8, 10], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10], Yes], [1, [0, 2, 4, 6, 8, 10], Yes], [2, [0, 2, 4, 6, 8, 10], Yes], [3, [0, 2, 4, 6, 8, 10], Yes], [4, [0, 2, 4, 6, 8, 10], Yes], [5, [0, 2, 4, 6, 8, 10], Yes], [6, [0, 2, 4, 6, 8, 10], Yes], [7, [0, 2, 4, 6, 8, 10], Yes], [8, [0, 2, 4, 6, 8, 10], Yes], [9, [0, 2, 4, 6, 8, 10], Yes], [10, [0, 2, 4, 6, 8, 10], Yes], [11, [0, 2, 4, 6, 8, 10], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0031","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"689d234ce6ec712c4eb9311882b28a2483149bdfb8fa147165a6cd4285c3c711","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141031,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12], Yes], [1, [0, 2, 4, 6, 8, 10, 12], Yes], [2, [0, 2, 4, 6, 8, 10, 12], Yes], [3, [0, 2, 4, 6, 8, 10, 12], Yes], [4, [0, 2, 4, 6, 8, 10, 12], Yes], [5, [0, 2, 4, 6, 8, 10, 12], Yes], [6, [0, 2, 4, 6, 8, 10, 12], Yes], [7, [0, 2, 4, 6, 8, 10, 12], Yes], [8, [0, 2, 4, 6, 8, 10, 12], Yes], [9, [0, 2, 4, 6, 8, 10, 12], Yes], [10, [0, 2, 4, 6, 8, 10, 12], Yes], [11, [0, 2, 4, 6, 8, 10, 12], Yes], [12, [0, 2, 4, 6, 8, 10, 12], Yes], [13, [0, 2, 4, 6, 8, 10, 12], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12], Yes], [1, [0, 2, 4, 6, 8, 10, 12], Yes], [2, [0, 2, 4, 6, 8, 10, 12], Yes], [3, [0, 2, 4, 6, 8, 10, 12], Yes], [4, [0, 2, 4, 6, 8, 10, 12], Yes], [5, [0, 2, 4, 6, 8, 10, 12], Yes], [6, [0, 2, 4, 6, 8, 10, 12], Yes], [7, [0, 2, 4, 6, 8, 10, 12], Yes], [8, [0, 2, 4, 6, 8, 10, 12], Yes], [9, [0, 2, 4, 6, 8, 10, 12], Yes], [10, [0, 2, 4, 6, 8, 10, 12], Yes], [11, [0, 2, 4, 6, 8, 10, 12], Yes], [12, [0, 2, 4, 6, 8, 10, 12], Yes], [13, [0, 2, 4, 6, 8, 10, 12], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0032","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04c003049bd45d75dfef3412ef2829b7fb0eea424b82497bc7420f76db4be02b","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141032,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0033","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32e1502691263b527faf1b2f3174aa3a6cb080e9dc2eb2e41b20957da0075e17","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141033,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0034","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d718ee584645f5d37a77fac4bd1d7ce427740bbf9c1494a232fb21bb73aa0c75","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141034,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0035","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ca2fbbf71f29ee7f6925e23a77939975026e010bac983598654f9bee2d09f98","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141035,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0036","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98419920bc3599d8bd31ced4a5eb966584a85d42419faaa7bd8d7e9401650217","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141036,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0037","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c33b8c1a6f3e83c74b468b19c64f41101958cdeebd0ed5617a3cbcc83108473","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141037,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0038","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"894f13d8160b19838cf009409f0299f47abbdc2a9162a5ae6e201900af311989","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141038,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0039","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf2428608ae6c453264af00b89cbe025034b6873e5365cb442e5a39f37fb1b4e","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141039,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0040","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebc881c05065066ddd2e4c567939689fa7961be6a072444338c858fb11f64f69","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141040,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0041","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc9eef1c315e2592ca6e3fe3c29fad7de4861d044802cdc010cb52247d5a33cd","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141041,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0042","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f645422af38b8b0aef32eb03ba4082461fff842d372c1a3263f08431515014ae","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141042,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0043","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39d7326e071b4630b791548eac09b3b539ac165d7cd572e7ea995974a5ba6332","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141043,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0044","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75830cacb5e6e94f6774e35862928b1812592362744d0a0ef491b5fe1c9e941d","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141044,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0045","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99e8afa476b1b7140c5e87a7c38cf6e39bc944108fbab62fa778f157d8721f4c","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141045,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0046","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5970f1c29fa293eee462a0d541063df66d61946a429fa9d7f7b7d9573ce52a42","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141046,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0047","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aae1a955283ba6d7044748b944334dec3c6ee9f0358534b7cb9416170b8c2afd","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141047,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0048","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a64af894c4fac40410bab6acf4080d0f79dc56d99727a54e2ca32ae9e31070cb","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141048,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0049","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26099160137ed36ef7227cc5d09c1ff0741b138ca32daa083eef546351ddc717","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141049,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0050","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f689da19de5c67197b94f3bd37832b3c87f951a36c5f27ba4b93d58c498fc473","family_id":"qf-mod45-kernel-normal-subgroup-audit","seed":202611141050,"semantic_template":"qf-mod45-kernel-normal-subgroup-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [50, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [51, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)","explanation":"Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [50, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [51, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)."},"skills":["The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0051","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae77f2176304c4edc163b6cd99ee2035940d7ee7ce91b80da0e2f4f060b9f789","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141051,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":0,\"b\":2,\"mapping\":[0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=0, b=2. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(0, 2, 0, 0, Yes, 2, Yes, [0, 2], [0, 2], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (0, 2, 0, 0, Yes, 2, Yes, [0, 2], [0, 2], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0052","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d307737a6f349e79b73d7f4cd4bb9764c060d8facee1ba45d0f647cfd0e3f5b4","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141052,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":1,\"b\":3,\"mapping\":[0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=1, b=3. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(1, 3, 1, 1, Yes, 2, Yes, [1, 3, 5], [1, 3, 5], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (1, 3, 1, 1, Yes, 2, Yes, [1, 3, 5], [1, 3, 5], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0053","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c6d56c5ec3a2441535ea994ff8134c568d6f71855fb0f519f3144b60c0da24e","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141053,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":2,\"b\":4,\"mapping\":[0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=2, b=4. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(2, 4, 0, 0, Yes, 2, Yes, [0, 2, 4, 6], [0, 2, 4, 6], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (2, 4, 0, 0, Yes, 2, Yes, [0, 2, 4, 6], [0, 2, 4, 6], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0054","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a1b5ea0a76d740e221b66dddbd7f3db47468386fdd7a582cfd7995ba1121086","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141054,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":3,\"b\":5,\"mapping\":[0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=3, b=5. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(3, 5, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9], [1, 3, 5, 7, 9], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (3, 5, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9], [1, 3, 5, 7, 9], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0055","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"480feca457d326b1d9e7f445ed210b130f22da25af8ae4ca9d647c39aed90984","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141055,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":4,\"b\":6,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=4, b=6. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(4, 6, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (4, 6, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0056","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5793345576ad6e48ad9a83bcab52e76429ac57edea25a6ef0ed797295966e8f9","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141056,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":5,\"b\":7,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=5, b=7. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(5, 7, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13], [1, 3, 5, 7, 9, 11, 13], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (5, 7, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13], [1, 3, 5, 7, 9, 11, 13], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0057","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e012fb581538dce3e296f278237c14d2aa4d4775e53df840ae8d9fcba182a6","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141057,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":6,\"b\":8,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=6, b=8. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(6, 8, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (6, 8, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0058","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e5e41b10748c62bd090f61956af1863c04bf34f04743fcc356c3e1893c45885","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141058,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":7,\"b\":9,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=7, b=9. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(7, 9, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17], [1, 3, 5, 7, 9, 11, 13, 15, 17], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (7, 9, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17], [1, 3, 5, 7, 9, 11, 13, 15, 17], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0059","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d63c65de529a796ec1ef92bdd07c2a4e2b1025ff6cf6dc92f48fc93746661404","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141059,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":8,\"b\":10,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=8, b=10. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(8, 10, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (8, 10, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0060","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edab006f4c9aff51f11cdba5ad07f4e16ad93447ec1ff89245851403f031136e","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141060,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":9,\"b\":11,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=9, b=11. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(9, 11, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (9, 11, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0061","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dba455a5c54d12433df6c307e87ed21be39541a9716aa50fae0cab09ccdc491b","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141061,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":10,\"b\":12,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=10, b=12. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(10, 12, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (10, 12, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0062","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"179b7681b333583a31d8ae0d87a4b4fd27ce02da73c515596ab7577b6bdaa858","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141062,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":11,\"b\":13,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=11, b=13. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(11, 13, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (11, 13, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0063","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c1f23accd83d39c10ba2fad54ee8c19eb8dda5584350caa3bcdbf742b4c7915","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141063,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":12,\"b\":14,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=12, b=14. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(12, 14, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (12, 14, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0064","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e6064c16a4f0459775bf34efafac9d9b63710ade3dd0b52c558cfc42622c356","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141064,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":13,\"b\":15,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=13, b=15. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(13, 15, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (13, 15, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0065","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b3192d4031ff60ca494fd4b46307e9d5fdf497ad607212bb36353dda150d828","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141065,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":14,\"b\":16,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=14, b=16. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(14, 16, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (14, 16, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0066","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fd2d30acb0d840830a9ce9b93bce2a9684d5c3ee11c91e230b3a886d884519a","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141066,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":15,\"b\":17,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=15, b=17. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(15, 17, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (15, 17, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0067","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bbd43d9a4b033936e78f17dbf61d63f1dd07707a8bc318cbf88e285bb931dc2","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141067,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":16,\"b\":18,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=16, b=18. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(16, 18, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (16, 18, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0068","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80d64982df8777a87546b2933eeb0a496016f8034ad15269c4ef82ce85c2005d","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141068,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":17,\"b\":19,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=17, b=19. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(17, 19, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (17, 19, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0069","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60a62f35ab135257d950869b017455c2442e21103d5de49de612a4dd515a4eba","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141069,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":18,\"b\":20,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=18, b=20. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(18, 20, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (18, 20, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0070","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"623485802bc08d21e1b2d18a0e8e86e79f0b73d4df1193d4181807629ee51983","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141070,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":19,\"b\":21,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=19, b=21. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(19, 21, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (19, 21, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0071","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c7f0a52c3a66e72f464c3fb43b2058847fb60be3562ec4879cd8ca82111b277","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141071,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":20,\"b\":22,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=20, b=22. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(20, 22, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (20, 22, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0072","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d3ed5688f6efa7a4113221c9631edb8bceae853c702f4ade3f09530d981ed84","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141072,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":21,\"b\":23,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=21, b=23. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(21, 23, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (21, 23, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0073","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52fcf620a24eba04369383314e050884aacc57b490cd37ea5854a0595ff9c492","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141073,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":22,\"b\":24,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=22, b=24. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(22, 24, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (22, 24, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0074","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86817fa8f09d62208835876fe2f26180135812bbcc89b602eacd358c0ca19d4c","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141074,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":23,\"b\":25,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=23, b=25. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(23, 25, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (23, 25, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0075","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c30f516c4ab1401d95ae400341955931558b7194906ccee80e267653c08945d0","family_id":"qf-mod45-kernel-fiber-coset-equivalence","seed":202611141075,"semantic_template":"qf-mod45-kernel-fiber-coset-equivalence/{\"a\":24,\"b\":26,\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]],\"target_identity\":0,\"target_table\":[[0,1],[1,0]]}"},"fields":{"title":"Image Equality Agrees with a Inverse Times b Membership in the Kernel","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=24, b=26. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(24, 26, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)","explanation":"Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (24, 26, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)."},"skills":["Image Equality Agrees with a Inverse Times b Membership in the Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0076","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb4912a01ba7f60b91b47f1a0b9ac0f19ceddd5139282dc7822a46ec99214f72","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141076,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2],\"source_identity\":0,\"source_table\":[[0,1,2],[1,2,0],[2,0,1]],\"target_identity\":0,\"target_table\":[[0,1,2],[1,2,0],[2,0,1]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], source identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], target identity=0, and total mapping=[0, 1, 2]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0077","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5a3f08ecd3feae56921b79170fb1e2c65811e6f4a991f988ce71ad906b2474a","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141077,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3],\"source_identity\":0,\"source_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],\"target_identity\":0,\"target_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], source identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], target identity=0, and total mapping=[0, 1, 2, 3]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0078","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57761f17223c8cb689dbeb72e4614fc99e6ba19de2705873ddbd092162409e4d","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141078,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4],\"source_identity\":0,\"source_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], source identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], target identity=0, and total mapping=[0, 1, 2, 3, 4]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0079","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc4d9e72a41ad8d2dae391febdd90ada301b782fdbba104bd270acb7feccf7d0","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141079,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], source identity=0, target table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0080","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98c148e23b5a211bd566f931e44204417db6b132285c8f0f27cbe435a1d085f4","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141080,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0081","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29fa21e6ca448278d2a4d75791a22059aa8ea0317d9a67746423de2dbe4752bb","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141081,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0082","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b8345d6dde030284d6f3cb8f87d1ed661fd0b7649d0b5b15d8132cf044aca32","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141082,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0083","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"036befc27722aca934d1fcf0f72d96a11548d5973e21f2a6b8a5eff972d372a6","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141083,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0084","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a1a6164c03b1d1f4821a57ec97e4ce13e71b4fc8827713ce4c542daad433386","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141084,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0085","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56eb64317c17d7951791bed7337688c10aa284866e671c5580b68d60dfd35118","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141085,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0086","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b21f906409e548c4a59f21f08b360f873a37786d609da1666e1dce36eb20a507","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141086,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0087","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30846a6874568b22951f0f1f8caf1233d0a267db3f0abf235573cb931cf413ba","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141087,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0088","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90c6f4630f7944a831e7e55b9a389d0213d4c72f9d42a09ffc2d67c506c67541","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141088,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0089","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2441de1096f0d34ebf1c508893e033e8674a153982dcfe72a5fd2e4596645a","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141089,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0090","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2874fdd87af9ff2c0695b8fe26333b6e673d77d67399273a73390bf99886e96c","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141090,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0091","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a175a6bbb3988ac647f27d1cd8f03c59eb1cc6e5419cd8bb1470b4d76472bc1","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141091,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0092","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e8b0de38046e1dffcd4f25594ae47a33592313d5e8e5106ea9bcd59051240c7","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141092,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0093","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7268b6013851dab4f10b63e94394d02a8abcdf9e90a5e2687ff6a7c4bd61876b","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141093,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0094","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f653bde899370def2d1cf50b228db2182b479dfd3c4402369cf1f5c072d7ee3","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141094,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0095","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"786d72d91fa96769465488142f13d9a90a24b42038e0408c719e4d7613484ce2","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141095,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0096","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f62a01accf471670fb53cac647a3e5f68dc20ac7182b902796225ebd848e5c33","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141096,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0097","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ea41c51e7a3185b2d3f1c9bad11669ba0bb4ea8c08acecec52e350a6b6a844a","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141097,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0098","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83127153c9bd0d33813d68a707e2f05a23e0509ea99b493cdeb630a3dc3da300","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141098,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0099","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d82e347c3b63fb8489b6bed54a45d4306258c2a101155f768fe2e5ca3d804fb4","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141099,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0100","batch_id":"batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5ab678a490b12e5976c97ac30bbd696179a84f53c00a6cc5732cbb755edc287","family_id":"qf-mod45-kernel-trivial-injective-boundary","seed":202611141100,"semantic_template":"qf-mod45-kernel-trivial-injective-boundary/{\"mapping\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]]}"},"fields":{"title":"A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem","stem":"Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).","answer":{"kind":"expression","value":"([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)","explanation":"Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)."},"skills":["A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}