{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0001","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce2f7d65e46b5db63151dae07f51a27ad00b20e52640a582e37df537b9346bc6","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151001,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(4, 2, [0, 1, 0, 1], Yes, null, [0, 2], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (4, 2, [0, 1, 0, 1], Yes, null, [0, 2], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0002","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff0f17d61a1880ed0e9662f1efd3f400d22656e9a507763bba605dd0a0fb5467","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151002,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(4, 4, [0, 1, 2, 3], Yes, null, [0], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (4, 4, [0, 1, 2, 3], Yes, null, [0], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0003","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65b69e3308949a2eb75cfd79590e0e94f45147432c772edb9b5fdfe8bf04c56b","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151003,"semantic_template":"qf-mod46-iso1-complete-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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(4, 4, [0, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (4, 4, [0, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0004","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b08170321920c6dc2f441326df76d0a2f6e2337e78885e588c9dbb828957e55d","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151004,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(5, 5, [0, 1, 2, 3, 4], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (5, 5, [0, 1, 2, 3, 4], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0005","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"add4994129c467e7cceb53d6e65a0ff4b9eb8bee27a7d1134cdd938d236635b9","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151005,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,2,4,1,3],\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 1, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(5, 5, [0, 2, 4, 1, 3], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (5, 5, [0, 2, 4, 1, 3], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0006","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"527aad521fc7a5eab0cd5ecf5191077bcf06801a879cce389d0c75e22f3ff97f","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151006,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,3,1,4,2],\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 1, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(5, 5, [0, 3, 1, 4, 2], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (5, 5, [0, 3, 1, 4, 2], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0007","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fae5d9d93da0a4edb89ba1cce86bed56decfbadf54ed5480ad36e5daf3ae0b03","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151007,"semantic_template":"qf-mod46-iso1-complete-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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(5, 5, [0, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (5, 5, [0, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0008","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1c2dbe3d60a41eaebc8c9b3b377baa9dcb36df6c2d8a59a5d9cd381e91d5518","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151008,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(6, 2, [0, 1, 0, 1, 0, 1], Yes, null, [0, 2, 4], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 2, [0, 1, 0, 1, 0, 1], Yes, null, [0, 2, 4], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0009","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"465426072bdfa37936d8918467669db707e14d5ec514989250bf3b7d57d0c6e4","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151009,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,1,2,0,1,2],\"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],[1,2,0],[2,0,1]]}"},"fields":{"title":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 1, 2, 0, 1, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(6, 3, [0, 1, 2, 0, 1, 2], Yes, null, [0, 3], [0, 1, 2], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 3, [0, 1, 2, 0, 1, 2], Yes, null, [0, 3], [0, 1, 2], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0010","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f8450d54d098ee8ca8fec8fbf79f412c50f8866a7861b4adf3428788cecae1b","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151010,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,2,1,0,2,1],\"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],[1,2,0],[2,0,1]]}"},"fields":{"title":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 2, 1, 0, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(6, 3, [0, 2, 1, 0, 2, 1], Yes, null, [0, 3], [0, 1, 2], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 3, [0, 2, 1, 0, 2, 1], Yes, null, [0, 3], [0, 1, 2], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0011","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e986d570186f9ffc696162360fed95a2891a88b13673b767fd9f468c6bf0bd2","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151011,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(6, 6, [0, 1, 2, 3, 4, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 6, [0, 1, 2, 3, 4, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0012","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d56b8e09ed7e7409f8ce1f5b8326e0fac1c3890d911c29a0cd417e2e3ddf81f3","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151012,"semantic_template":"qf-mod46-iso1-complete-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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(6, 6, [0, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (6, 6, [0, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0013","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb368bc6bb523c5464bd3332804477e72a7c0114f08a4256e14f7590eb26c949","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151013,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(7, 7, [0, 1, 2, 3, 4, 5, 6], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 1, 2, 3, 4, 5, 6], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0014","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"635d282d3c8ef594a55603b8143603be54414a58e240b7738ff4892aef0af9c6","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151014,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,2,4,6,1,3,5],\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 1, 3, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(7, 7, [0, 2, 4, 6, 1, 3, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 2, 4, 6, 1, 3, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0015","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"808a56bbf5063ed4bc5268b2033c982143af033cfdaae05dc268ec877a155a6e","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151015,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,3,6,2,5,1,4],\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 6, 2, 5, 1, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(7, 7, [0, 3, 6, 2, 5, 1, 4], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 3, 6, 2, 5, 1, 4], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0016","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59a20bae77d6af0f48e6f9c4f917e705838d09e921db164f64d5dcd2652efd7e","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151016,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,4,1,5,2,6,3],\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 4, 1, 5, 2, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(7, 7, [0, 4, 1, 5, 2, 6, 3], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 4, 1, 5, 2, 6, 3], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0017","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1d17c9f82b9734a6ac25a5c9d40a8b09b663d42b37aaaadd7c1d1afc842fc01","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151017,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,5,3,1,6,4,2],\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 3, 1, 6, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(7, 7, [0, 5, 3, 1, 6, 4, 2], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 5, 3, 1, 6, 4, 2], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0018","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a77a1c48720edff27239f09a028d4e582d468d50c05ffd4bd8cbab123e6fa466","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151018,"semantic_template":"qf-mod46-iso1-complete-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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(7, 7, [0, 6, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (7, 7, [0, 6, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0019","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"925ae684aff14972b1eaf00072bc77dedde595737def63b94f1a15422253eacd","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151019,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(8, 2, [0, 1, 0, 1, 0, 1, 0, 1], Yes, null, [0, 2, 4, 6], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 2, [0, 1, 0, 1, 0, 1, 0, 1], Yes, null, [0, 2, 4, 6], [0, 1], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0020","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c2e44911a94bf15915e319c1acbdf58c00eac8c8e4c60a4c2497a4e49ab4c9d","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151020,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,1,2,3,0,1,2,3],\"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],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(8, 4, [0, 1, 2, 3, 0, 1, 2, 3], Yes, null, [0, 4], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 4, [0, 1, 2, 3, 0, 1, 2, 3], Yes, null, [0, 4], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0021","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c4781aabba48afa83091cc9ecdc8821d6c871e74fe06981c524dc434825cd44","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151021,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,3,2,1,0,3,2,1],\"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],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1, 0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(8, 4, [0, 3, 2, 1, 0, 3, 2, 1], Yes, null, [0, 4], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 4, [0, 3, 2, 1, 0, 3, 2, 1], Yes, null, [0, 4], [0, 1, 2, 3], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0022","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a87b3730be99f4e329d537f544bc494ac23f2e9c4a82ca562174fd331ed8503a","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151022,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6, 7]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(8, 8, [0, 1, 2, 3, 4, 5, 6, 7], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 8, [0, 1, 2, 3, 4, 5, 6, 7], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0023","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89182db8287999fe8360ea197f03747c0177b7c60c9f9022e9c124ad31e7ac57","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151023,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,3,6,1,4,7,2,5],\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 6, 1, 4, 7, 2, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(8, 8, [0, 3, 6, 1, 4, 7, 2, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 8, [0, 3, 6, 1, 4, 7, 2, 5], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0024","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a1422670cbab8b0feb10384bf8123442b508c9a4df9f984503443c7a998e24b","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151024,"semantic_template":"qf-mod46-iso1-complete-qualification/{\"mapping\":[0,5,2,7,4,1,6,3],\"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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 2, 7, 4, 1, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(8, 8, [0, 5, 2, 7, 4, 1, 6, 3], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 8, [0, 5, 2, 7, 4, 1, 6, 3], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0025","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b516e8af9f6b3ddafca1c19da5a38d9e973cecbb6bc0d68e881a9611e2909cf0","family_id":"qf-mod46-iso1-complete-qualification","seed":202611151025,"semantic_template":"qf-mod46-iso1-complete-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":"Homomorphism, Kernel and Image Qualification Precede First Isomorphism","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]], 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]], identity=0, and the aligned total mapping=[0, 7, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate).","answer":{"kind":"expression","value":"(8, 8, [0, 7, 6, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)","explanation":"Check the complete product law, retaining the first failed pair or null, the full identity preimage and full image. A total map need not be a homomorphism. Kernel normality and image subgroup flags are checked properties, not inferred from the theorem name. The exact result, retaining the requested order, is (8, 8, [0, 7, 6, 5, 4, 3, 2, 1], Yes, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)."},"skills":["Homomorphism, Kernel and Image Qualification Precede First Isomorphism"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0026","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e86ec0faec12f9e9771cc261c280298224032255178bef724a0081dc9c02d2e","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151026,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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 Induced Map Must Agree on Every Kernel-Coset Representative","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0, 2], [[0, 2], [1, 3]], [[0, [0, 2], [0], Yes], [1, [1, 3], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 2], [[0, 2], [1, 3]], [[0, [0, 2], [0], Yes], [1, [1, 3], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0027","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb6fd58a7aa2a4a9c5db97e6f5e877c91e0a955c399d0c72b333fc0f353f35ba","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151027,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0028","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0677e2f4e3bacda1ca467a0f16fb3e0c638d3ed64e6b29ebcc988634fbc4485c","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151028,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [2], Yes], [3, [3], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [2], Yes], [3, [3], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0029","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d7a3fb6dfbbf9f1336e997ccf9f45df431a2340af279b5425f4e027061a0805","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151029,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0030","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fab491689f0d3ab7584732ab1af0f64e7561af38609d9ac43d3771bb846d0e39","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151030,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,2,4,1,3],\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 1, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [2], [4], Yes], [3, [3], [1], Yes], [4, [4], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [2], [4], Yes], [3, [3], [1], Yes], [4, [4], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0031","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f464ee0bbe6449718c995294b48b47c23b9c8b54d4ae465f39cb44455f82e34e","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151031,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,3,1,4,2],\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 1, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [1], Yes], [3, [3], [4], Yes], [4, [4], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [1], Yes], [3, [3], [4], Yes], [4, [4], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0032","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a56b285a4aafc36cdc2675a45a4492df4fc0cd5a8ef7e3e8508aa4fa46410baa","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151032,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [4], Yes], [2, [2], [3], Yes], [3, [3], [2], Yes], [4, [4], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], Yes], [1, [1], [4], Yes], [2, [2], [3], Yes], [3, [3], [2], Yes], [4, [4], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0033","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be1644d48eda829b0762c14b487f58819fe3d25848da96c4680a19dec14ebc4f","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151033,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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 Induced Map Must Agree on Every Kernel-Coset Representative","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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0, 2, 4], [[0, 2, 4], [1, 3, 5]], [[0, [0, 2, 4], [0], Yes], [1, [1, 3, 5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 2, 4], [[0, 2, 4], [1, 3, 5]], [[0, [0, 2, 4], [0], Yes], [1, [1, 3, 5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0034","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"552d6e22a0e7c98628d38f2f67f07b7e09a2bba76b73290bf159cd9109508bb5","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151034,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,1,2,0,1,2],\"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],[1,2,0],[2,0,1]]}"},"fields":{"title":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 1, 2, 0, 1, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], Yes], [1, [1, 4], [1], Yes], [2, [2, 5], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], Yes], [1, [1, 4], [1], Yes], [2, [2, 5], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0035","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98d5a2d33e6b4e7f2a385b1e812b1bc9ed0ba009258fb80f56a4e3f0f64f74cc","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151035,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,2,1,0,2,1],\"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],[1,2,0],[2,0,1]]}"},"fields":{"title":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 2, 1, 0, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], Yes], [1, [1, 4], [2], Yes], [2, [2, 5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], Yes], [1, [1, 4], [2], Yes], [2, [2, 5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0036","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb973e01e56a150d87c7f194de31cd3aa42e916313ad5fb1b78d4543a675f43","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151036,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0037","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf5aa0ddac1fc7f272f1437952f28c5496e87821c5e9a0321d03f1f235234e4b","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151037,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [4], Yes], [3, [3], [3], Yes], [4, [4], [2], Yes], [5, [5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [4], Yes], [3, [3], [3], Yes], [4, [4], [2], Yes], [5, [5], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0038","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49913d3837680d5d73071822618ffe2d644334fcbf2f6fc7b7ab5ba5ef4c80c4","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151038,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0039","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"408c7cd2103b03cc08177e0e64b57def564dad9beb1dbbb2d479da47450e9867","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151039,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,2,4,6,1,3,5],\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 1, 3, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [2], [4], Yes], [3, [3], [6], Yes], [4, [4], [1], Yes], [5, [5], [3], Yes], [6, [6], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [2], [4], Yes], [3, [3], [6], Yes], [4, [4], [1], Yes], [5, [5], [3], Yes], [6, [6], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0040","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd6400b8497d11c19f679873724cb543c99450da62a863cfa0cc030ae3161895","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151040,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,3,6,2,5,1,4],\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 6, 2, 5, 1, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [6], Yes], [3, [3], [2], Yes], [4, [4], [5], Yes], [5, [5], [1], Yes], [6, [6], [4], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [6], Yes], [3, [3], [2], Yes], [4, [4], [5], Yes], [5, [5], [1], Yes], [6, [6], [4], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0041","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"819743bd8e296872cbe99de77632cc29743dd6c94047e376c4819e27dd637bde","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151041,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,4,1,5,2,6,3],\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 4, 1, 5, 2, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [4], Yes], [2, [2], [1], Yes], [3, [3], [5], Yes], [4, [4], [2], Yes], [5, [5], [6], Yes], [6, [6], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [4], Yes], [2, [2], [1], Yes], [3, [3], [5], Yes], [4, [4], [2], Yes], [5, [5], [6], Yes], [6, [6], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0042","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec8f5dae0ee3baa4efa7a5201ec34ef873770fdfa8e263f089fd1819c94fd4ae","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151042,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,5,3,1,6,4,2],\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 3, 1, 6, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [3], Yes], [3, [3], [1], Yes], [4, [4], [6], Yes], [5, [5], [4], Yes], [6, [6], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [3], Yes], [3, [3], [1], Yes], [4, [4], [6], Yes], [5, [5], [4], Yes], [6, [6], [2], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0043","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13746cded92b21ea29901ed70d4deb431dd59b8c8579e983e778334ecd2538be","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151043,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [6], Yes], [2, [2], [5], Yes], [3, [3], [4], Yes], [4, [4], [3], Yes], [5, [5], [2], Yes], [6, [6], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], Yes], [1, [1], [6], Yes], [2, [2], [5], Yes], [3, [3], [4], Yes], [4, [4], [3], Yes], [5, [5], [2], Yes], [6, [6], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0044","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d973e83e637debb57131832cf2f926105538d45b6fad60686043f818edaa9b8","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151044,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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 Induced Map Must Agree on Every Kernel-Coset Representative","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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], [[0, [0, 2, 4, 6], [0], Yes], [1, [1, 3, 5, 7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], [[0, [0, 2, 4, 6], [0], Yes], [1, [1, 3, 5, 7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0045","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4b1b10c36c51d5ce41fa485528e51346e2893cd80ec218de7030cee8e0f72c8","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151045,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,1,2,3,0,1,2,3],\"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],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], Yes], [1, [1, 5], [1], Yes], [2, [2, 6], [2], Yes], [3, [3, 7], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], Yes], [1, [1, 5], [1], Yes], [2, [2, 6], [2], Yes], [3, [3, 7], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0046","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8e2788dabf489d5b540cc8988aeb253ad09c17c7c50c7ca326cf37db502cbe2","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151046,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,3,2,1,0,3,2,1],\"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],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1, 0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], Yes], [1, [1, 5], [3], Yes], [2, [2, 6], [2], Yes], [3, [3, 7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], Yes], [1, [1, 5], [3], Yes], [2, [2, 6], [2], Yes], [3, [3, 7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0047","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88785ab73731904aef65913ffbcb4dd1da851ce300ceaf9ddd783d019fc84080","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151047,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6, 7]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes], [7, [7], [7], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [1], Yes], [2, [2], [2], Yes], [3, [3], [3], Yes], [4, [4], [4], Yes], [5, [5], [5], Yes], [6, [6], [6], Yes], [7, [7], [7], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0048","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85721fe25cf157fb900d3dfd30fed6e8f2ae2e9f4663e3cff9c56adf2041c1a1","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151048,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,3,6,1,4,7,2,5],\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 6, 1, 4, 7, 2, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [6], Yes], [3, [3], [1], Yes], [4, [4], [4], Yes], [5, [5], [7], Yes], [6, [6], [2], Yes], [7, [7], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [3], Yes], [2, [2], [6], Yes], [3, [3], [1], Yes], [4, [4], [4], Yes], [5, [5], [7], Yes], [6, [6], [2], Yes], [7, [7], [5], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0049","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"674e5d3c036c97a1bf8cac72a57560708d3ebf1b5aee35f53097584e2778a9ce","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151049,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"mapping\":[0,5,2,7,4,1,6,3],\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 2, 7, 4, 1, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [2], Yes], [3, [3], [7], Yes], [4, [4], [4], Yes], [5, [5], [1], Yes], [6, [6], [6], Yes], [7, [7], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [5], Yes], [2, [2], [2], Yes], [3, [3], [7], Yes], [4, [4], [4], Yes], [5, [5], [1], Yes], [6, [6], [6], Yes], [7, [7], [3], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0050","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"116206c0f2fc19efad251b1394dc1fcb4146a62117d83d8e7c693d99408327d3","family_id":"qf-mod46-iso1-induced-well-definedness","seed":202611151050,"semantic_template":"qf-mod46-iso1-induced-well-definedness/{\"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":"The Induced Map Must Agree on Every Kernel-Coset Representative","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]], 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]], identity=0, and the aligned total mapping=[0, 7, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).","answer":{"kind":"expression","value":"([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [7], Yes], [2, [2], [6], Yes], [3, [3], [5], Yes], [4, [4], [4], Yes], [5, [5], [3], Yes], [6, [6], [2], Yes], [7, [7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)","explanation":"Require a genuine homomorphism and a normal kernel. Build the entire coset carrier and retain every coset image set. Each must be a singleton before bar_f(g*kernel)=f(g) is well-defined. Quotient elements are cosets, not chosen representatives. The exact result, retaining the requested order, is ([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], Yes], [1, [1], [7], Yes], [2, [2], [6], Yes], [3, [3], [5], Yes], [4, [4], [4], Yes], [5, [5], [3], Yes], [6, [6], [2], Yes], [7, [7], [1], Yes]], Yes, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)."},"skills":["The Induced Map Must Agree on Every Kernel-Coset Representative"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0051","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b29541a2265e29d60e059ca2cac8a43eff49d8bfd62ec55c1898f7fca6fc1dd9","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151051,"semantic_template":"qf-mod46-iso1-induced-isomorphism-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 Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0, 2], [1, 3]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 2], [1, 3]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0052","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da3b58987f7d5acf5b8c2d8308c43713edec8f71fc390c1b593c9a8f4fd644ea","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151052,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 1, 2, 3], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 1, 2, 3], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0053","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d59689159c4b57068f994ed83d0ffacab8372464c22e4f162487e60469a7ff13","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151053,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0054","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df662b0258447ac1dbf4971a6678abc269817d792f4b262fabbd59cdd696c67d","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151054,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[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]], [0, 1, 2, 3, 4], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 0, 0, Yes], [2, 4, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 0, 0, Yes], [3, 3, 1, 1, Yes], [3, 4, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 0, 0, Yes], [4, 2, 1, 1, Yes], [4, 3, 2, 2, Yes], [4, 4, 3, 3, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[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]], [0, 1, 2, 3, 4], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 0, 0, Yes], [2, 4, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 0, 0, Yes], [3, 3, 1, 1, Yes], [3, 4, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 0, 0, Yes], [4, 2, 1, 1, Yes], [4, 3, 2, 2, Yes], [4, 4, 3, 3, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0055","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb53ab45c4029ee662a62121c1d68b25157d35d9b15a9fd30512ba3241b2f60","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151055,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,2,4,1,3],\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 1, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[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]], [0, 2, 4, 1, 3], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 1, 1, Yes], [0, 4, 3, 3, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 1, 1, Yes], [1, 3, 3, 3, Yes], [1, 4, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 1, 1, Yes], [2, 2, 3, 3, Yes], [2, 3, 0, 0, Yes], [2, 4, 2, 2, Yes], [3, 0, 1, 1, Yes], [3, 1, 3, 3, Yes], [3, 2, 0, 0, Yes], [3, 3, 2, 2, Yes], [3, 4, 4, 4, Yes], [4, 0, 3, 3, Yes], [4, 1, 0, 0, Yes], [4, 2, 2, 2, Yes], [4, 3, 4, 4, Yes], [4, 4, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[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]], [0, 2, 4, 1, 3], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 1, 1, Yes], [0, 4, 3, 3, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 1, 1, Yes], [1, 3, 3, 3, Yes], [1, 4, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 1, 1, Yes], [2, 2, 3, 3, Yes], [2, 3, 0, 0, Yes], [2, 4, 2, 2, Yes], [3, 0, 1, 1, Yes], [3, 1, 3, 3, Yes], [3, 2, 0, 0, Yes], [3, 3, 2, 2, Yes], [3, 4, 4, 4, Yes], [4, 0, 3, 3, Yes], [4, 1, 0, 0, Yes], [4, 2, 2, 2, Yes], [4, 3, 4, 4, Yes], [4, 4, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0056","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cc9460093c0b92bdd7d6f9d60a602d042d4a7738f99b3bfbb97b162c8a26001","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151056,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,3,1,4,2],\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 1, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[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]], [0, 3, 1, 4, 2], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 1, 1, Yes], [0, 3, 4, 4, Yes], [0, 4, 2, 2, Yes], [1, 0, 3, 3, Yes], [1, 1, 1, 1, Yes], [1, 2, 4, 4, Yes], [1, 3, 2, 2, Yes], [1, 4, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 4, 4, Yes], [2, 2, 2, 2, Yes], [2, 3, 0, 0, Yes], [2, 4, 3, 3, Yes], [3, 0, 4, 4, Yes], [3, 1, 2, 2, Yes], [3, 2, 0, 0, Yes], [3, 3, 3, 3, Yes], [3, 4, 1, 1, Yes], [4, 0, 2, 2, Yes], [4, 1, 0, 0, Yes], [4, 2, 3, 3, Yes], [4, 3, 1, 1, Yes], [4, 4, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[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]], [0, 3, 1, 4, 2], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 1, 1, Yes], [0, 3, 4, 4, Yes], [0, 4, 2, 2, Yes], [1, 0, 3, 3, Yes], [1, 1, 1, 1, Yes], [1, 2, 4, 4, Yes], [1, 3, 2, 2, Yes], [1, 4, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 4, 4, Yes], [2, 2, 2, 2, Yes], [2, 3, 0, 0, Yes], [2, 4, 3, 3, Yes], [3, 0, 4, 4, Yes], [3, 1, 2, 2, Yes], [3, 2, 0, 0, Yes], [3, 3, 3, 3, Yes], [3, 4, 1, 1, Yes], [4, 0, 2, 2, Yes], [4, 1, 0, 0, Yes], [4, 2, 3, 3, Yes], [4, 3, 1, 1, Yes], [4, 4, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0057","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f33d4238570eb85f5fa31f9afcc714617e6965c334f60da56561c9c5220dba7","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151057,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[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]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4]], [0, 1, 2, 3, 4], [[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]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0058","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"740ba3d873bc67cba9225b4ef1c65c30b0c97c23fd13ac2e026f699be0da0eb8","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151058,"semantic_template":"qf-mod46-iso1-induced-isomorphism-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 Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0, 2, 4], [1, 3, 5]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 2, 4], [1, 3, 5]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0059","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"100b3659fe870252fc2fc2a224b3dd5a9b7db4d3a05ace9447af2e8d81e01724","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151059,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,1,2,0,1,2],\"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],[1,2,0],[2,0,1]]}"},"fields":{"title":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 1, 2, 0, 1, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0, 3], [1, 4], [2, 5]], [0, 1, 2], [[0, 1, 2], [1, 2, 0], [2, 0, 1]], [0, 1, 2], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 0, 0, Yes], [2, 2, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 3], [1, 4], [2, 5]], [0, 1, 2], [[0, 1, 2], [1, 2, 0], [2, 0, 1]], [0, 1, 2], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 0, 0, Yes], [2, 2, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0060","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a07695d44713467816e90bc0409069646599b8f8ff50a1be1d5d7790e35823b8","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151060,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,2,1,0,2,1],\"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],[1,2,0],[2,0,1]]}"},"fields":{"title":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, and the aligned total mapping=[0, 2, 1, 0, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0, 3], [1, 4], [2, 5]], [0, 1, 2], [[0, 1, 2], [1, 2, 0], [2, 0, 1]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 3], [1, 4], [2, 5]], [0, 1, 2], [[0, 1, 2], [1, 2, 0], [2, 0, 1]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0061","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4698f16f319a25e292e7e394074b2b1ef267ac9cd0baa7a9218ed83be545a183","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151061,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[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]], [0, 1, 2, 3, 4, 5], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 0, 0, Yes], [2, 5, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 0, 0, Yes], [3, 4, 1, 1, Yes], [3, 5, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 0, 0, Yes], [4, 3, 1, 1, Yes], [4, 4, 2, 2, Yes], [4, 5, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 0, 0, Yes], [5, 2, 1, 1, Yes], [5, 3, 2, 2, Yes], [5, 4, 3, 3, Yes], [5, 5, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[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]], [0, 1, 2, 3, 4, 5], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 0, 0, Yes], [2, 5, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 0, 0, Yes], [3, 4, 1, 1, Yes], [3, 5, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 0, 0, Yes], [4, 3, 1, 1, Yes], [4, 4, 2, 2, Yes], [4, 5, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 0, 0, Yes], [5, 2, 1, 1, Yes], [5, 3, 2, 2, Yes], [5, 4, 3, 3, Yes], [5, 5, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0062","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e84979a1fa813c5415f034c800bf63c0a94dd7c57e0d2a24ed49a6277e4f3f0","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151062,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[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]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[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]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0063","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f66f8ae7cfe197897e6e845dc9e9e21028ba1fa2ad43fe87f1402df28f7b084f","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151063,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 1, 2, 3, 4, 5, 6], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 0, 0, Yes], [2, 6, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 0, 0, Yes], [3, 5, 1, 1, Yes], [3, 6, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 0, 0, Yes], [4, 4, 1, 1, Yes], [4, 5, 2, 2, Yes], [4, 6, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 0, 0, Yes], [5, 3, 1, 1, Yes], [5, 4, 2, 2, Yes], [5, 5, 3, 3, Yes], [5, 6, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 0, 0, Yes], [6, 2, 1, 1, Yes], [6, 3, 2, 2, Yes], [6, 4, 3, 3, Yes], [6, 5, 4, 4, Yes], [6, 6, 5, 5, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 1, 2, 3, 4, 5, 6], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 0, 0, Yes], [2, 6, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 0, 0, Yes], [3, 5, 1, 1, Yes], [3, 6, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 0, 0, Yes], [4, 4, 1, 1, Yes], [4, 5, 2, 2, Yes], [4, 6, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 0, 0, Yes], [5, 3, 1, 1, Yes], [5, 4, 2, 2, Yes], [5, 5, 3, 3, Yes], [5, 6, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 0, 0, Yes], [6, 2, 1, 1, Yes], [6, 3, 2, 2, Yes], [6, 4, 3, 3, Yes], [6, 5, 4, 4, Yes], [6, 6, 5, 5, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0064","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b6929e02d710835ac5e22df9b795091721e1b6514a736cb39596f0863b08028","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151064,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,2,4,6,1,3,5],\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 1, 3, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 2, 4, 6, 1, 3, 5], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 6, 6, Yes], [0, 4, 1, 1, Yes], [0, 5, 3, 3, Yes], [0, 6, 5, 5, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 6, 6, Yes], [1, 3, 1, 1, Yes], [1, 4, 3, 3, Yes], [1, 5, 5, 5, Yes], [1, 6, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 6, 6, Yes], [2, 2, 1, 1, Yes], [2, 3, 3, 3, Yes], [2, 4, 5, 5, Yes], [2, 5, 0, 0, Yes], [2, 6, 2, 2, Yes], [3, 0, 6, 6, Yes], [3, 1, 1, 1, Yes], [3, 2, 3, 3, Yes], [3, 3, 5, 5, Yes], [3, 4, 0, 0, Yes], [3, 5, 2, 2, Yes], [3, 6, 4, 4, Yes], [4, 0, 1, 1, Yes], [4, 1, 3, 3, Yes], [4, 2, 5, 5, Yes], [4, 3, 0, 0, Yes], [4, 4, 2, 2, Yes], [4, 5, 4, 4, Yes], [4, 6, 6, 6, Yes], [5, 0, 3, 3, Yes], [5, 1, 5, 5, Yes], [5, 2, 0, 0, Yes], [5, 3, 2, 2, Yes], [5, 4, 4, 4, Yes], [5, 5, 6, 6, Yes], [5, 6, 1, 1, Yes], [6, 0, 5, 5, Yes], [6, 1, 0, 0, Yes], [6, 2, 2, 2, Yes], [6, 3, 4, 4, Yes], [6, 4, 6, 6, Yes], [6, 5, 1, 1, Yes], [6, 6, 3, 3, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 2, 4, 6, 1, 3, 5], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 6, 6, Yes], [0, 4, 1, 1, Yes], [0, 5, 3, 3, Yes], [0, 6, 5, 5, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 6, 6, Yes], [1, 3, 1, 1, Yes], [1, 4, 3, 3, Yes], [1, 5, 5, 5, Yes], [1, 6, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 6, 6, Yes], [2, 2, 1, 1, Yes], [2, 3, 3, 3, Yes], [2, 4, 5, 5, Yes], [2, 5, 0, 0, Yes], [2, 6, 2, 2, Yes], [3, 0, 6, 6, Yes], [3, 1, 1, 1, Yes], [3, 2, 3, 3, Yes], [3, 3, 5, 5, Yes], [3, 4, 0, 0, Yes], [3, 5, 2, 2, Yes], [3, 6, 4, 4, Yes], [4, 0, 1, 1, Yes], [4, 1, 3, 3, Yes], [4, 2, 5, 5, Yes], [4, 3, 0, 0, Yes], [4, 4, 2, 2, Yes], [4, 5, 4, 4, Yes], [4, 6, 6, 6, Yes], [5, 0, 3, 3, Yes], [5, 1, 5, 5, Yes], [5, 2, 0, 0, Yes], [5, 3, 2, 2, Yes], [5, 4, 4, 4, Yes], [5, 5, 6, 6, Yes], [5, 6, 1, 1, Yes], [6, 0, 5, 5, Yes], [6, 1, 0, 0, Yes], [6, 2, 2, 2, Yes], [6, 3, 4, 4, Yes], [6, 4, 6, 6, Yes], [6, 5, 1, 1, Yes], [6, 6, 3, 3, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0065","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81117a7dae4deda7373b6305cae8420cc92411bdde6b52a6f96bd202c308784c","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151065,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,3,6,2,5,1,4],\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 6, 2, 5, 1, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 3, 6, 2, 5, 1, 4], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 6, 6, Yes], [0, 3, 2, 2, Yes], [0, 4, 5, 5, Yes], [0, 5, 1, 1, Yes], [0, 6, 4, 4, Yes], [1, 0, 3, 3, Yes], [1, 1, 6, 6, Yes], [1, 2, 2, 2, Yes], [1, 3, 5, 5, Yes], [1, 4, 1, 1, Yes], [1, 5, 4, 4, Yes], [1, 6, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 2, 2, Yes], [2, 2, 5, 5, Yes], [2, 3, 1, 1, Yes], [2, 4, 4, 4, Yes], [2, 5, 0, 0, Yes], [2, 6, 3, 3, Yes], [3, 0, 2, 2, Yes], [3, 1, 5, 5, Yes], [3, 2, 1, 1, Yes], [3, 3, 4, 4, Yes], [3, 4, 0, 0, Yes], [3, 5, 3, 3, Yes], [3, 6, 6, 6, Yes], [4, 0, 5, 5, Yes], [4, 1, 1, 1, Yes], [4, 2, 4, 4, Yes], [4, 3, 0, 0, Yes], [4, 4, 3, 3, Yes], [4, 5, 6, 6, Yes], [4, 6, 2, 2, Yes], [5, 0, 1, 1, Yes], [5, 1, 4, 4, Yes], [5, 2, 0, 0, Yes], [5, 3, 3, 3, Yes], [5, 4, 6, 6, Yes], [5, 5, 2, 2, Yes], [5, 6, 5, 5, Yes], [6, 0, 4, 4, Yes], [6, 1, 0, 0, Yes], [6, 2, 3, 3, Yes], [6, 3, 6, 6, Yes], [6, 4, 2, 2, Yes], [6, 5, 5, 5, Yes], [6, 6, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 3, 6, 2, 5, 1, 4], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 6, 6, Yes], [0, 3, 2, 2, Yes], [0, 4, 5, 5, Yes], [0, 5, 1, 1, Yes], [0, 6, 4, 4, Yes], [1, 0, 3, 3, Yes], [1, 1, 6, 6, Yes], [1, 2, 2, 2, Yes], [1, 3, 5, 5, Yes], [1, 4, 1, 1, Yes], [1, 5, 4, 4, Yes], [1, 6, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 2, 2, Yes], [2, 2, 5, 5, Yes], [2, 3, 1, 1, Yes], [2, 4, 4, 4, Yes], [2, 5, 0, 0, Yes], [2, 6, 3, 3, Yes], [3, 0, 2, 2, Yes], [3, 1, 5, 5, Yes], [3, 2, 1, 1, Yes], [3, 3, 4, 4, Yes], [3, 4, 0, 0, Yes], [3, 5, 3, 3, Yes], [3, 6, 6, 6, Yes], [4, 0, 5, 5, Yes], [4, 1, 1, 1, Yes], [4, 2, 4, 4, Yes], [4, 3, 0, 0, Yes], [4, 4, 3, 3, Yes], [4, 5, 6, 6, Yes], [4, 6, 2, 2, Yes], [5, 0, 1, 1, Yes], [5, 1, 4, 4, Yes], [5, 2, 0, 0, Yes], [5, 3, 3, 3, Yes], [5, 4, 6, 6, Yes], [5, 5, 2, 2, Yes], [5, 6, 5, 5, Yes], [6, 0, 4, 4, Yes], [6, 1, 0, 0, Yes], [6, 2, 3, 3, Yes], [6, 3, 6, 6, Yes], [6, 4, 2, 2, Yes], [6, 5, 5, 5, Yes], [6, 6, 1, 1, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0066","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae9cfc470a6c9d83a8eb9972ae9c1127032feb71a82a070f13dcecf9f656bd4e","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151066,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,4,1,5,2,6,3],\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 4, 1, 5, 2, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 4, 1, 5, 2, 6, 3], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 1, 1, Yes], [0, 3, 5, 5, Yes], [0, 4, 2, 2, Yes], [0, 5, 6, 6, Yes], [0, 6, 3, 3, Yes], [1, 0, 4, 4, Yes], [1, 1, 1, 1, Yes], [1, 2, 5, 5, Yes], [1, 3, 2, 2, Yes], [1, 4, 6, 6, Yes], [1, 5, 3, 3, Yes], [1, 6, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 5, 5, Yes], [2, 2, 2, 2, Yes], [2, 3, 6, 6, Yes], [2, 4, 3, 3, Yes], [2, 5, 0, 0, Yes], [2, 6, 4, 4, Yes], [3, 0, 5, 5, Yes], [3, 1, 2, 2, Yes], [3, 2, 6, 6, Yes], [3, 3, 3, 3, Yes], [3, 4, 0, 0, Yes], [3, 5, 4, 4, Yes], [3, 6, 1, 1, Yes], [4, 0, 2, 2, Yes], [4, 1, 6, 6, Yes], [4, 2, 3, 3, Yes], [4, 3, 0, 0, Yes], [4, 4, 4, 4, Yes], [4, 5, 1, 1, Yes], [4, 6, 5, 5, Yes], [5, 0, 6, 6, Yes], [5, 1, 3, 3, Yes], [5, 2, 0, 0, Yes], [5, 3, 4, 4, Yes], [5, 4, 1, 1, Yes], [5, 5, 5, 5, Yes], [5, 6, 2, 2, Yes], [6, 0, 3, 3, Yes], [6, 1, 0, 0, Yes], [6, 2, 4, 4, Yes], [6, 3, 1, 1, Yes], [6, 4, 5, 5, Yes], [6, 5, 2, 2, Yes], [6, 6, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 4, 1, 5, 2, 6, 3], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 1, 1, Yes], [0, 3, 5, 5, Yes], [0, 4, 2, 2, Yes], [0, 5, 6, 6, Yes], [0, 6, 3, 3, Yes], [1, 0, 4, 4, Yes], [1, 1, 1, 1, Yes], [1, 2, 5, 5, Yes], [1, 3, 2, 2, Yes], [1, 4, 6, 6, Yes], [1, 5, 3, 3, Yes], [1, 6, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 5, 5, Yes], [2, 2, 2, 2, Yes], [2, 3, 6, 6, Yes], [2, 4, 3, 3, Yes], [2, 5, 0, 0, Yes], [2, 6, 4, 4, Yes], [3, 0, 5, 5, Yes], [3, 1, 2, 2, Yes], [3, 2, 6, 6, Yes], [3, 3, 3, 3, Yes], [3, 4, 0, 0, Yes], [3, 5, 4, 4, Yes], [3, 6, 1, 1, Yes], [4, 0, 2, 2, Yes], [4, 1, 6, 6, Yes], [4, 2, 3, 3, Yes], [4, 3, 0, 0, Yes], [4, 4, 4, 4, Yes], [4, 5, 1, 1, Yes], [4, 6, 5, 5, Yes], [5, 0, 6, 6, Yes], [5, 1, 3, 3, Yes], [5, 2, 0, 0, Yes], [5, 3, 4, 4, Yes], [5, 4, 1, 1, Yes], [5, 5, 5, 5, Yes], [5, 6, 2, 2, Yes], [6, 0, 3, 3, Yes], [6, 1, 0, 0, Yes], [6, 2, 4, 4, Yes], [6, 3, 1, 1, Yes], [6, 4, 5, 5, Yes], [6, 5, 2, 2, Yes], [6, 6, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0067","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b9ea88c4f57f8e57a919b5f1ce5176762c2e067313a85d8cf154e839020e22f","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151067,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,5,3,1,6,4,2],\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 3, 1, 6, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 5, 3, 1, 6, 4, 2], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 3, 3, Yes], [0, 3, 1, 1, Yes], [0, 4, 6, 6, Yes], [0, 5, 4, 4, Yes], [0, 6, 2, 2, Yes], [1, 0, 5, 5, Yes], [1, 1, 3, 3, Yes], [1, 2, 1, 1, Yes], [1, 3, 6, 6, Yes], [1, 4, 4, 4, Yes], [1, 5, 2, 2, Yes], [1, 6, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 1, 1, Yes], [2, 2, 6, 6, Yes], [2, 3, 4, 4, Yes], [2, 4, 2, 2, Yes], [2, 5, 0, 0, Yes], [2, 6, 5, 5, Yes], [3, 0, 1, 1, Yes], [3, 1, 6, 6, Yes], [3, 2, 4, 4, Yes], [3, 3, 2, 2, Yes], [3, 4, 0, 0, Yes], [3, 5, 5, 5, Yes], [3, 6, 3, 3, Yes], [4, 0, 6, 6, Yes], [4, 1, 4, 4, Yes], [4, 2, 2, 2, Yes], [4, 3, 0, 0, Yes], [4, 4, 5, 5, Yes], [4, 5, 3, 3, Yes], [4, 6, 1, 1, Yes], [5, 0, 4, 4, Yes], [5, 1, 2, 2, Yes], [5, 2, 0, 0, Yes], [5, 3, 5, 5, Yes], [5, 4, 3, 3, Yes], [5, 5, 1, 1, Yes], [5, 6, 6, 6, Yes], [6, 0, 2, 2, Yes], [6, 1, 0, 0, Yes], [6, 2, 5, 5, Yes], [6, 3, 3, 3, Yes], [6, 4, 1, 1, Yes], [6, 5, 6, 6, Yes], [6, 6, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [0, 5, 3, 1, 6, 4, 2], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 3, 3, Yes], [0, 3, 1, 1, Yes], [0, 4, 6, 6, Yes], [0, 5, 4, 4, Yes], [0, 6, 2, 2, Yes], [1, 0, 5, 5, Yes], [1, 1, 3, 3, Yes], [1, 2, 1, 1, Yes], [1, 3, 6, 6, Yes], [1, 4, 4, 4, Yes], [1, 5, 2, 2, Yes], [1, 6, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 1, 1, Yes], [2, 2, 6, 6, Yes], [2, 3, 4, 4, Yes], [2, 4, 2, 2, Yes], [2, 5, 0, 0, Yes], [2, 6, 5, 5, Yes], [3, 0, 1, 1, Yes], [3, 1, 6, 6, Yes], [3, 2, 4, 4, Yes], [3, 3, 2, 2, Yes], [3, 4, 0, 0, Yes], [3, 5, 5, 5, Yes], [3, 6, 3, 3, Yes], [4, 0, 6, 6, Yes], [4, 1, 4, 4, Yes], [4, 2, 2, 2, Yes], [4, 3, 0, 0, Yes], [4, 4, 5, 5, Yes], [4, 5, 3, 3, Yes], [4, 6, 1, 1, Yes], [5, 0, 4, 4, Yes], [5, 1, 2, 2, Yes], [5, 2, 0, 0, Yes], [5, 3, 5, 5, Yes], [5, 4, 3, 3, Yes], [5, 5, 1, 1, Yes], [5, 6, 6, 6, Yes], [6, 0, 2, 2, Yes], [6, 1, 0, 0, Yes], [6, 2, 5, 5, Yes], [6, 3, 3, 3, Yes], [6, 4, 1, 1, Yes], [6, 5, 6, 6, Yes], [6, 6, 4, 4, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0068","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7af4c6b3f6b0f6e618447d445b83f572188c5aecf16ee4220f13bb29121b1ba6","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151068,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], [[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]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0069","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ec40b44c59ee20a3e1018fdd9facdca78b7dc09ef1bb5ba74584d7884662cd6","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151069,"semantic_template":"qf-mod46-iso1-induced-isomorphism-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 Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], identity=0, target table=[[0, 1], [1, 0]], identity=0, and the aligned total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0, 2, 4, 6], [1, 3, 5, 7]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 2, 4, 6], [1, 3, 5, 7]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0070","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8921c9c8c8b5de4685881826636e44ebd32bea4751c10cd4e45e2a5335eab00","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151070,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,1,2,3,0,1,2,3],\"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],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 0, 1, 2, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0, 4], [1, 5], [2, 6], [3, 7]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 1, 2, 3], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 4], [1, 5], [2, 6], [3, 7]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 1, 2, 3], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0071","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67abce367b4c8cf5c1720636bf29231321be1c5ebebb4754b3ba03630cb3d5fc","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151071,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,3,2,1,0,3,2,1],\"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],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 3, 2, 1, 0, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0, 4], [1, 5], [2, 6], [3, 7]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0, 4], [1, 5], [2, 6], [3, 7]], [0, 1, 2, 3], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0072","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98f9189cdeba9834f360feee2f9857131d7f2df5eef59dc4064c85f2424cea8e","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151072,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 1, 2, 3, 4, 5, 6, 7]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 1, 2, 3, 4, 5, 6, 7], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [0, 7, 7, 7, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 7, 7, Yes], [1, 7, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 7, 7, Yes], [2, 6, 0, 0, Yes], [2, 7, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 7, 7, Yes], [3, 5, 0, 0, Yes], [3, 6, 1, 1, Yes], [3, 7, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 7, 7, Yes], [4, 4, 0, 0, Yes], [4, 5, 1, 1, Yes], [4, 6, 2, 2, Yes], [4, 7, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 7, 7, Yes], [5, 3, 0, 0, Yes], [5, 4, 1, 1, Yes], [5, 5, 2, 2, Yes], [5, 6, 3, 3, Yes], [5, 7, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 7, 7, Yes], [6, 2, 0, 0, Yes], [6, 3, 1, 1, Yes], [6, 4, 2, 2, Yes], [6, 5, 3, 3, Yes], [6, 6, 4, 4, Yes], [6, 7, 5, 5, Yes], [7, 0, 7, 7, Yes], [7, 1, 0, 0, Yes], [7, 2, 1, 1, Yes], [7, 3, 2, 2, Yes], [7, 4, 3, 3, Yes], [7, 5, 4, 4, Yes], [7, 6, 5, 5, Yes], [7, 7, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 1, 2, 3, 4, 5, 6, 7], [[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [0, 7, 7, 7, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 7, 7, Yes], [1, 7, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 7, 7, Yes], [2, 6, 0, 0, Yes], [2, 7, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 7, 7, Yes], [3, 5, 0, 0, Yes], [3, 6, 1, 1, Yes], [3, 7, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 7, 7, Yes], [4, 4, 0, 0, Yes], [4, 5, 1, 1, Yes], [4, 6, 2, 2, Yes], [4, 7, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 7, 7, Yes], [5, 3, 0, 0, Yes], [5, 4, 1, 1, Yes], [5, 5, 2, 2, Yes], [5, 6, 3, 3, Yes], [5, 7, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 7, 7, Yes], [6, 2, 0, 0, Yes], [6, 3, 1, 1, Yes], [6, 4, 2, 2, Yes], [6, 5, 3, 3, Yes], [6, 6, 4, 4, Yes], [6, 7, 5, 5, Yes], [7, 0, 7, 7, Yes], [7, 1, 0, 0, Yes], [7, 2, 1, 1, Yes], [7, 3, 2, 2, Yes], [7, 4, 3, 3, Yes], [7, 5, 4, 4, Yes], [7, 6, 5, 5, Yes], [7, 7, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0073","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ae2cc26e7f97cb897fa121c4fa2660bf454d420ec60b86f5fa4ffec1d64038e","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151073,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,3,6,1,4,7,2,5],\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 3, 6, 1, 4, 7, 2, 5]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 3, 6, 1, 4, 7, 2, 5], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 6, 6, Yes], [0, 3, 1, 1, Yes], [0, 4, 4, 4, Yes], [0, 5, 7, 7, Yes], [0, 6, 2, 2, Yes], [0, 7, 5, 5, Yes], [1, 0, 3, 3, Yes], [1, 1, 6, 6, Yes], [1, 2, 1, 1, Yes], [1, 3, 4, 4, Yes], [1, 4, 7, 7, Yes], [1, 5, 2, 2, Yes], [1, 6, 5, 5, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 1, 1, Yes], [2, 2, 4, 4, Yes], [2, 3, 7, 7, Yes], [2, 4, 2, 2, Yes], [2, 5, 5, 5, Yes], [2, 6, 0, 0, Yes], [2, 7, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 4, 4, Yes], [3, 2, 7, 7, Yes], [3, 3, 2, 2, Yes], [3, 4, 5, 5, Yes], [3, 5, 0, 0, Yes], [3, 6, 3, 3, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 7, 7, Yes], [4, 2, 2, 2, Yes], [4, 3, 5, 5, Yes], [4, 4, 0, 0, Yes], [4, 5, 3, 3, Yes], [4, 6, 6, 6, Yes], [4, 7, 1, 1, Yes], [5, 0, 7, 7, Yes], [5, 1, 2, 2, Yes], [5, 2, 5, 5, Yes], [5, 3, 0, 0, Yes], [5, 4, 3, 3, Yes], [5, 5, 6, 6, Yes], [5, 6, 1, 1, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 5, 5, Yes], [6, 2, 0, 0, Yes], [6, 3, 3, 3, Yes], [6, 4, 6, 6, Yes], [6, 5, 1, 1, Yes], [6, 6, 4, 4, Yes], [6, 7, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 0, 0, Yes], [7, 2, 3, 3, Yes], [7, 3, 6, 6, Yes], [7, 4, 1, 1, Yes], [7, 5, 4, 4, Yes], [7, 6, 7, 7, Yes], [7, 7, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 3, 6, 1, 4, 7, 2, 5], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 6, 6, Yes], [0, 3, 1, 1, Yes], [0, 4, 4, 4, Yes], [0, 5, 7, 7, Yes], [0, 6, 2, 2, Yes], [0, 7, 5, 5, Yes], [1, 0, 3, 3, Yes], [1, 1, 6, 6, Yes], [1, 2, 1, 1, Yes], [1, 3, 4, 4, Yes], [1, 4, 7, 7, Yes], [1, 5, 2, 2, Yes], [1, 6, 5, 5, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 1, 1, Yes], [2, 2, 4, 4, Yes], [2, 3, 7, 7, Yes], [2, 4, 2, 2, Yes], [2, 5, 5, 5, Yes], [2, 6, 0, 0, Yes], [2, 7, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 4, 4, Yes], [3, 2, 7, 7, Yes], [3, 3, 2, 2, Yes], [3, 4, 5, 5, Yes], [3, 5, 0, 0, Yes], [3, 6, 3, 3, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 7, 7, Yes], [4, 2, 2, 2, Yes], [4, 3, 5, 5, Yes], [4, 4, 0, 0, Yes], [4, 5, 3, 3, Yes], [4, 6, 6, 6, Yes], [4, 7, 1, 1, Yes], [5, 0, 7, 7, Yes], [5, 1, 2, 2, Yes], [5, 2, 5, 5, Yes], [5, 3, 0, 0, Yes], [5, 4, 3, 3, Yes], [5, 5, 6, 6, Yes], [5, 6, 1, 1, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 5, 5, Yes], [6, 2, 0, 0, Yes], [6, 3, 3, 3, Yes], [6, 4, 6, 6, Yes], [6, 5, 1, 1, Yes], [6, 6, 4, 4, Yes], [6, 7, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 0, 0, Yes], [7, 2, 3, 3, Yes], [7, 3, 6, 6, Yes], [7, 4, 1, 1, Yes], [7, 5, 4, 4, Yes], [7, 6, 7, 7, Yes], [7, 7, 2, 2, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0074","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"419b177c30fbd3c99de1e7eb56cc7c1f46ef81db71a86d03bbd4e674a305dee9","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151074,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"mapping\":[0,5,2,7,4,1,6,3],\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 5, 2, 7, 4, 1, 6, 3]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 5, 2, 7, 4, 1, 6, 3], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 2, 2, Yes], [0, 3, 7, 7, Yes], [0, 4, 4, 4, Yes], [0, 5, 1, 1, Yes], [0, 6, 6, 6, Yes], [0, 7, 3, 3, Yes], [1, 0, 5, 5, Yes], [1, 1, 2, 2, Yes], [1, 2, 7, 7, Yes], [1, 3, 4, 4, Yes], [1, 4, 1, 1, Yes], [1, 5, 6, 6, Yes], [1, 6, 3, 3, Yes], [1, 7, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 7, 7, Yes], [2, 2, 4, 4, Yes], [2, 3, 1, 1, Yes], [2, 4, 6, 6, Yes], [2, 5, 3, 3, Yes], [2, 6, 0, 0, Yes], [2, 7, 5, 5, Yes], [3, 0, 7, 7, Yes], [3, 1, 4, 4, Yes], [3, 2, 1, 1, Yes], [3, 3, 6, 6, Yes], [3, 4, 3, 3, Yes], [3, 5, 0, 0, Yes], [3, 6, 5, 5, Yes], [3, 7, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 1, 1, Yes], [4, 2, 6, 6, Yes], [4, 3, 3, 3, Yes], [4, 4, 0, 0, Yes], [4, 5, 5, 5, Yes], [4, 6, 2, 2, Yes], [4, 7, 7, 7, Yes], [5, 0, 1, 1, Yes], [5, 1, 6, 6, Yes], [5, 2, 3, 3, Yes], [5, 3, 0, 0, Yes], [5, 4, 5, 5, Yes], [5, 5, 2, 2, Yes], [5, 6, 7, 7, Yes], [5, 7, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 3, 3, Yes], [6, 2, 0, 0, Yes], [6, 3, 5, 5, Yes], [6, 4, 2, 2, Yes], [6, 5, 7, 7, Yes], [6, 6, 4, 4, Yes], [6, 7, 1, 1, Yes], [7, 0, 3, 3, Yes], [7, 1, 0, 0, Yes], [7, 2, 5, 5, Yes], [7, 3, 2, 2, Yes], [7, 4, 7, 7, Yes], [7, 5, 4, 4, Yes], [7, 6, 1, 1, Yes], [7, 7, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [0, 5, 2, 7, 4, 1, 6, 3], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 2, 2, Yes], [0, 3, 7, 7, Yes], [0, 4, 4, 4, Yes], [0, 5, 1, 1, Yes], [0, 6, 6, 6, Yes], [0, 7, 3, 3, Yes], [1, 0, 5, 5, Yes], [1, 1, 2, 2, Yes], [1, 2, 7, 7, Yes], [1, 3, 4, 4, Yes], [1, 4, 1, 1, Yes], [1, 5, 6, 6, Yes], [1, 6, 3, 3, Yes], [1, 7, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 7, 7, Yes], [2, 2, 4, 4, Yes], [2, 3, 1, 1, Yes], [2, 4, 6, 6, Yes], [2, 5, 3, 3, Yes], [2, 6, 0, 0, Yes], [2, 7, 5, 5, Yes], [3, 0, 7, 7, Yes], [3, 1, 4, 4, Yes], [3, 2, 1, 1, Yes], [3, 3, 6, 6, Yes], [3, 4, 3, 3, Yes], [3, 5, 0, 0, Yes], [3, 6, 5, 5, Yes], [3, 7, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 1, 1, Yes], [4, 2, 6, 6, Yes], [4, 3, 3, 3, Yes], [4, 4, 0, 0, Yes], [4, 5, 5, 5, Yes], [4, 6, 2, 2, Yes], [4, 7, 7, 7, Yes], [5, 0, 1, 1, Yes], [5, 1, 6, 6, Yes], [5, 2, 3, 3, Yes], [5, 3, 0, 0, Yes], [5, 4, 5, 5, Yes], [5, 5, 2, 2, Yes], [5, 6, 7, 7, Yes], [5, 7, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 3, 3, Yes], [6, 2, 0, 0, Yes], [6, 3, 5, 5, Yes], [6, 4, 2, 2, Yes], [6, 5, 7, 7, Yes], [6, 6, 4, 4, Yes], [6, 7, 1, 1, Yes], [7, 0, 3, 3, Yes], [7, 1, 0, 0, Yes], [7, 2, 5, 5, Yes], [7, 3, 2, 2, Yes], [7, 4, 7, 7, Yes], [7, 5, 4, 4, Yes], [7, 6, 1, 1, Yes], [7, 7, 6, 6, Yes]], Yes, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0075","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"987a1762365ac36dd9606218ca6d9e650503cd06459bdc334496b2c7f09dc18d","family_id":"qf-mod46-iso1-induced-isomorphism-audit","seed":202611151075,"semantic_template":"qf-mod46-iso1-induced-isomorphism-audit/{\"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":"The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit","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]], 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]], identity=0, and the aligned total mapping=[0, 7, 6, 5, 4, 3, 2, 1]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate).","answer":{"kind":"expression","value":"([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)","explanation":"Build the quotient table and induced images in canonical coset order. Check every multiplication pair, injectivity and surjectivity to the actual image. Equal cardinalities alone do not certify an isomorphism, and the whole codomain is not silently substituted for the image. The exact result, retaining the requested order, is ([[0], [1], [2], [3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], [[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]], [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, Yes, Yes, Yes, induced_map_is_a_bijective_group_homomorphism_onto_image)."},"skills":["The Quotient-to-Image Map Requires a Complete Homomorphism and Bijection Audit"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0076","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a74a6624f8695424ac67bf324762d67f4b0114691691024370bdf7427d827d3f","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151076,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2],\"source_identity\":0,\"source_table\":[[0,1],[1,0]],\"target_identity\":0,\"target_table\":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","stem":"Use complete source Cayley table=[[0, 1], [1, 0]], identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, and the aligned total mapping=[0, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2], 4, 2, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2], 4, 2, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0077","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bf4689aad22d91fd3c1925d6ce31e55cae47501d22193a26ba67f1b5c968219","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151077,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4],\"source_identity\":0,\"source_table\":[[0,1,2],[1,2,0],[2,0,1]],\"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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","stem":"Use complete source Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]], identity=0, and the aligned total mapping=[0, 2, 4]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4], 6, 3, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4], 6, 3, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0078","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"855a31eeb128e40d01435502ca1a196f8d54f9e9429e113113a08bddfcab7c61","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151078,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6],\"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\":[[0,1,2,3,4,5,6,7],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","stem":"Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6], 8, 4, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6], 8, 4, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0079","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9f1ef54918e10d4240e651528abbc0dfd87e051991cfcb56d842b0ed9e835fe","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151079,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8],\"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\":[[0,1,2,3,4,5,6,7,8,9],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8], 10, 5, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8], 10, 5, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0080","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b8c1ec9093762c448c89f3b4f66da4b06175af6e81dca9a8fd6485ec85f7ed1","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151080,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10], 12, 6, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10], 12, 6, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0081","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b1737516d5892c743a6baebe4c89ab30c01d4e009b7b83e0e1426e46cacca54","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151081,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12], 14, 7, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12], 14, 7, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0082","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74d1960bd7d34043225b3ca55d365c454448e2c0b370989490db8804f2bb31f9","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151082,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14], 16, 8, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14], 16, 8, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0083","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a8e351ad8fdb85df3218e33688dcd921b08caa7360be273b9b98dc3c3f727e6","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151083,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16], 18, 9, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16], 18, 9, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0084","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aa36db12412ae28e5cd079c42796426d09b338c23f4dbf39220d3557c72f38e","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151084,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 20, 10, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 20, 10, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0085","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f07307e78c999284f46a72dec179312d7f934abdeb3c655315c8c5a0a34c3bd8","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151085,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 22, 11, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 22, 11, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0086","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abbead0318c398b5bfabab7566ad1ef0d70e689f0c28a09aa19e31abf9003cf4","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151086,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 24, 12, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 24, 12, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0087","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76261e7c994ac8973c9c2fa331f4c1495c528d1f90646b7a0abbef250e6c2b44","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151087,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[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":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 26, 13, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 26, 13, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0088","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a09fa1313a4a0c6787912f2a6daabbf8271d95207b962a085051852638cfaaa","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151088,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 28, 14, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 28, 14, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0089","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab5924d278941be0697ee31b961bb432bc2b777906e661772f298672785923a4","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151089,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 30, 15, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 30, 15, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0090","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55ce432de521795ea92d1139cb7538d411d120644a06854d3c78c78b692a173d","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151090,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 32, 16, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 16, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0091","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bef0e1ccdd3a9a2287b334dddff2fb10a4e55c7bd1e094e67fa50f51163f229a","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151091,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 34, 17, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 17, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0092","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"135018f162764b54a079ff661b237dc21bfd8d470468a1cb15bc37d2d0255707","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151092,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 36, 18, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 18, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0093","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70a2dd682e826c4a50a3534d7c26b2a1785897fa0e0f0ecf49be96874bb4dd1c","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151093,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","answer":{"kind":"expression","value":"([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 38, 19, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 19, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0094","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21e69b4196bd00839fbde6ee9eb286347da6f3fa8e97e94a9d9332b7b377d29a","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151094,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","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, 20, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 20, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0095","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4caa905725f16c6b2a91854442f76578f47d284b537ce519bcf55c8bbd86e17a","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151095,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","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, 21, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 21, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0096","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e324ba6a20e76bb102c5fcdca18e62038403d0c4078e2598ab194e860d3138e","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151096,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","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, 22, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 22, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0097","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6592782b4ec2fb570116c3c75571721a804f759c83983e9bbc4c47e226d0a47","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151097,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","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, 23, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 23, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0098","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dd7bfe440fdbfec86733bf0cfc65ef7e257366c56d0bf49a6c55e047df14a76","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151098,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","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, 24, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 24, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0099","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62acedec4fd03b83f350cfa479667e93b827fbaea940ee032b937e2e6d740edb","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151099,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 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]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","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, 25, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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, 25, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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-20261115-r01-first-isomorphism-theorem-finite-groups-0100","batch_id":"batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d6f117be5e2e970de6320ca9572c33d68e44478c41cc2748102aae22f7e42f3","family_id":"qf-mod46-iso1-image-codomain-boundary","seed":202611151100,"semantic_template":"qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50],\"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\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[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]]}"},"fields":{"title":"First Isomorphism Reaches the Image, Not Automatically the Whole Codomain","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]], 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, 27, 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]], identity=0, and the aligned total mapping=[0, 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]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).","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], 52, 26, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)","explanation":"Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. 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], 52, 26, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."},"skills":["First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"],"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."}}