# 有限群第一同構定理基礎 · 脈衝 R01

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations · 2026-10-15 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

[欄序與來源綁定資料](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(4, 2, [0, 1, 0, 1], 是, null, [0, 2], [0, 1], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(4, 2, [0, 1, 0, 1], 是, null, [0, 2], [0, 1], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0001/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 2. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,1,2,3]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(4, 4, [0, 1, 2, 3], 是, null, [0], [0, 1, 2, 3], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(4, 4, [0, 1, 2, 3], 是, null, [0], [0, 1, 2, 3], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0002/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 3. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,3,2,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(4, 4, [0, 3, 2, 1], 是, null, [0], [0, 1, 2, 3], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(4, 4, [0, 3, 2, 1], 是, null, [0], [0, 1, 2, 3], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0003/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 4. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,1,2,3,4]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(5, 5, [0, 1, 2, 3, 4], 是, null, [0], [0, 1, 2, 3, 4], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(5, 5, [0, 1, 2, 3, 4], 是, null, [0], [0, 1, 2, 3, 4], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0004/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 5. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,2,4,1,3]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(5, 5, [0, 2, 4, 1, 3], 是, null, [0], [0, 1, 2, 3, 4], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(5, 5, [0, 2, 4, 1, 3], 是, null, [0], [0, 1, 2, 3, 4], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0005/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 6. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,3,1,4,2]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(5, 5, [0, 3, 1, 4, 2], 是, null, [0], [0, 1, 2, 3, 4], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(5, 5, [0, 3, 1, 4, 2], 是, null, [0], [0, 1, 2, 3, 4], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0006/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 7. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,4,3,2,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(5, 5, [0, 4, 3, 2, 1], 是, null, [0], [0, 1, 2, 3, 4], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(5, 5, [0, 4, 3, 2, 1], 是, null, [0], [0, 1, 2, 3, 4], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0007/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 8. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(6, 2, [0, 1, 0, 1, 0, 1], 是, null, [0, 2, 4], [0, 1], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(6, 2, [0, 1, 0, 1, 0, 1], 是, null, [0, 2, 4], [0, 1], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0008/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 9. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"target_identity":0,"mapping":[0,1,2,0,1,2]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(6, 3, [0, 1, 2, 0, 1, 2], 是, null, [0, 3], [0, 1, 2], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(6, 3, [0, 1, 2, 0, 1, 2], 是, null, [0, 3], [0, 1, 2], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0009/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 10. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"target_identity":0,"mapping":[0,2,1,0,2,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(6, 3, [0, 2, 1, 0, 2, 1], 是, null, [0, 3], [0, 1, 2], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(6, 3, [0, 2, 1, 0, 2, 1], 是, null, [0, 3], [0, 1, 2], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0010/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 11. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,1,2,3,4,5]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(6, 6, [0, 1, 2, 3, 4, 5], 是, null, [0], [0, 1, 2, 3, 4, 5], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(6, 6, [0, 1, 2, 3, 4, 5], 是, null, [0], [0, 1, 2, 3, 4, 5], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0011/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 12. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,5,4,3,2,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(6, 6, [0, 5, 4, 3, 2, 1], 是, null, [0], [0, 1, 2, 3, 4, 5], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(6, 6, [0, 5, 4, 3, 2, 1], 是, null, [0], [0, 1, 2, 3, 4, 5], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0012/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 13. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,1,2,3,4,5,6]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(7, 7, [0, 1, 2, 3, 4, 5, 6], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(7, 7, [0, 1, 2, 3, 4, 5, 6], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0013/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 14. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,2,4,6,1,3,5]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(7, 7, [0, 2, 4, 6, 1, 3, 5], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(7, 7, [0, 2, 4, 6, 1, 3, 5], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0014/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 15. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,3,6,2,5,1,4]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(7, 7, [0, 3, 6, 2, 5, 1, 4], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(7, 7, [0, 3, 6, 2, 5, 1, 4], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0015/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 16. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,4,1,5,2,6,3]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(7, 7, [0, 4, 1, 5, 2, 6, 3], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(7, 7, [0, 4, 1, 5, 2, 6, 3], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0016/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 17. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,5,3,1,6,4,2]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(7, 7, [0, 5, 3, 1, 6, 4, 2], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(7, 7, [0, 5, 3, 1, 6, 4, 2], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0017/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 18. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,6,5,4,3,2,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(7, 7, [0, 6, 5, 4, 3, 2, 1], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(7, 7, [0, 6, 5, 4, 3, 2, 1], 是, null, [0], [0, 1, 2, 3, 4, 5, 6], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0018/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 19. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(8, 2, [0, 1, 0, 1, 0, 1, 0, 1], 是, null, [0, 2, 4, 6], [0, 1], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(8, 2, [0, 1, 0, 1, 0, 1, 0, 1], 是, null, [0, 2, 4, 6], [0, 1], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0019/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 20. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,1,2,3,0,1,2,3]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(8, 4, [0, 1, 2, 3, 0, 1, 2, 3], 是, null, [0, 4], [0, 1, 2, 3], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(8, 4, [0, 1, 2, 3, 0, 1, 2, 3], 是, null, [0, 4], [0, 1, 2, 3], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0020/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 21. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,3,2,1,0,3,2,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(8, 4, [0, 3, 2, 1, 0, 3, 2, 1], 是, null, [0, 4], [0, 1, 2, 3], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(8, 4, [0, 3, 2, 1, 0, 3, 2, 1], 是, null, [0, 4], [0, 1, 2, 3], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0021/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 22. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(8, 8, [0, 1, 2, 3, 4, 5, 6, 7], 是, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(8, 8, [0, 1, 2, 3, 4, 5, 6, 7], 是, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0022/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 23. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,3,6,1,4,7,2,5]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(8, 8, [0, 3, 6, 1, 4, 7, 2, 5], 是, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(8, 8, [0, 3, 6, 1, 4, 7, 2, 5], 是, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0023/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 24. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,5,2,7,4,1,6,3]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(8, 8, [0, 5, 2, 7, 4, 1, 6, 3], 是, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(8, 8, [0, 5, 2, 7, 4, 1, 6, 3], 是, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0024/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 25. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-complete-qualification；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,7,6,5,4,3,2,1]}。請依 [source_order, target_order, mapping, homomorphism, first_failed_pair_or_null, kernel, image, normal_kernel, image_subgroup, certificate] 欄序回答。

查看答案與解釋(8, 8, [0, 7, 6, 5, 4, 3, 2, 1], 是, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)

Wave6第九層核對=(8, 8, [0, 7, 6, 5, 4, 3, 2, 1], 是, null, [0], [0, 1, 2, 3, 4, 5, 6, 7], 是, 是, homomorphism_kernel_image_and_quotient_qualifications_precede_the_theorem)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0025/)

有限群第一同構定理 · qf-mod46-iso1-complete-qualification · candidate





## 26. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0, 2], [[0, 2], [1, 3]], [[0, [0, 2], [0], 是], [1, [1, 3], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0, 2], [[0, 2], [1, 3]], [[0, [0, 2], [0], 是], [1, [1, 3], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0026/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 27. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,1,2,3]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0027/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 28. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,3,2,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3]], [[0, [0], [0], 是], [1, [1], [3], 是], [2, [2], [2], 是], [3, [3], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3]], [[0, [0], [0], 是], [1, [1], [3], 是], [2, [2], [2], 是], [3, [3], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0028/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 29. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,1,2,3,4]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0029/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 30. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,2,4,1,3]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], 是], [1, [1], [2], 是], [2, [2], [4], 是], [3, [3], [1], 是], [4, [4], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], 是], [1, [1], [2], 是], [2, [2], [4], 是], [3, [3], [1], 是], [4, [4], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0030/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 31. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,3,1,4,2]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], 是], [1, [1], [3], 是], [2, [2], [1], 是], [3, [3], [4], 是], [4, [4], [2], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], 是], [1, [1], [3], 是], [2, [2], [1], 是], [3, [3], [4], 是], [4, [4], [2], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0031/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 32. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,4,3,2,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], 是], [1, [1], [4], 是], [2, [2], [3], 是], [3, [3], [2], 是], [4, [4], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4]], [[0, [0], [0], 是], [1, [1], [4], 是], [2, [2], [3], 是], [3, [3], [2], 是], [4, [4], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0032/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 33. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0, 2, 4], [[0, 2, 4], [1, 3, 5]], [[0, [0, 2, 4], [0], 是], [1, [1, 3, 5], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0, 2, 4], [[0, 2, 4], [1, 3, 5]], [[0, [0, 2, 4], [0], 是], [1, [1, 3, 5], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0033/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 34. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"target_identity":0,"mapping":[0,1,2,0,1,2]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], 是], [1, [1, 4], [1], 是], [2, [2, 5], [2], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], 是], [1, [1, 4], [1], 是], [2, [2, 5], [2], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0034/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 35. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"target_identity":0,"mapping":[0,2,1,0,2,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], 是], [1, [1, 4], [2], 是], [2, [2, 5], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0, 3], [[0, 3], [1, 4], [2, 5]], [[0, [0, 3], [0], 是], [1, [1, 4], [2], 是], [2, [2, 5], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0035/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 36. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,1,2,3,4,5]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0036/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 37. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,5,4,3,2,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], 是], [1, [1], [5], 是], [2, [2], [4], 是], [3, [3], [3], 是], [4, [4], [2], 是], [5, [5], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5]], [[0, [0], [0], 是], [1, [1], [5], 是], [2, [2], [4], 是], [3, [3], [3], 是], [4, [4], [2], 是], [5, [5], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0037/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 38. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,1,2,3,4,5,6]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0038/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 39. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,2,4,6,1,3,5]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [2], 是], [2, [2], [4], 是], [3, [3], [6], 是], [4, [4], [1], 是], [5, [5], [3], 是], [6, [6], [5], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [2], 是], [2, [2], [4], 是], [3, [3], [6], 是], [4, [4], [1], 是], [5, [5], [3], 是], [6, [6], [5], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0039/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 40. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,3,6,2,5,1,4]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [3], 是], [2, [2], [6], 是], [3, [3], [2], 是], [4, [4], [5], 是], [5, [5], [1], 是], [6, [6], [4], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [3], 是], [2, [2], [6], 是], [3, [3], [2], 是], [4, [4], [5], 是], [5, [5], [1], 是], [6, [6], [4], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0040/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 41. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,4,1,5,2,6,3]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [4], 是], [2, [2], [1], 是], [3, [3], [5], 是], [4, [4], [2], 是], [5, [5], [6], 是], [6, [6], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [4], 是], [2, [2], [1], 是], [3, [3], [5], 是], [4, [4], [2], 是], [5, [5], [6], 是], [6, [6], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0041/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 42. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,5,3,1,6,4,2]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [5], 是], [2, [2], [3], 是], [3, [3], [1], 是], [4, [4], [6], 是], [5, [5], [4], 是], [6, [6], [2], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [5], 是], [2, [2], [3], 是], [3, [3], [1], 是], [4, [4], [6], 是], [5, [5], [4], 是], [6, [6], [2], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0042/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 43. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,6,5,4,3,2,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [6], 是], [2, [2], [5], 是], [3, [3], [4], 是], [4, [4], [3], 是], [5, [5], [2], 是], [6, [6], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6]], [[0, [0], [0], 是], [1, [1], [6], 是], [2, [2], [5], 是], [3, [3], [4], 是], [4, [4], [3], 是], [5, [5], [2], 是], [6, [6], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0043/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 44. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], [[0, [0, 2, 4, 6], [0], 是], [1, [1, 3, 5, 7], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], [[0, [0, 2, 4, 6], [0], 是], [1, [1, 3, 5, 7], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0044/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 45. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,1,2,3,0,1,2,3]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], 是], [1, [1, 5], [1], 是], [2, [2, 6], [2], 是], [3, [3, 7], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], 是], [1, [1, 5], [1], 是], [2, [2, 6], [2], 是], [3, [3, 7], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0045/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 46. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,3,2,1,0,3,2,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], 是], [1, [1, 5], [3], 是], [2, [2, 6], [2], 是], [3, [3, 7], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], [[0, [0, 4], [0], 是], [1, [1, 5], [3], 是], [2, [2, 6], [2], 是], [3, [3, 7], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0046/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 47. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是], [6, [6], [6], 是], [7, [7], [7], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0047/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 48. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,3,6,1,4,7,2,5]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], 是], [1, [1], [3], 是], [2, [2], [6], 是], [3, [3], [1], 是], [4, [4], [4], 是], [5, [5], [7], 是], [6, [6], [2], 是], [7, [7], [5], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], 是], [1, [1], [3], 是], [2, [2], [6], 是], [3, [3], [1], 是], [4, [4], [4], 是], [5, [5], [7], 是], [6, [6], [2], 是], [7, [7], [5], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0048/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 49. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,5,2,7,4,1,6,3]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], 是], [1, [1], [5], 是], [2, [2], [2], 是], [3, [3], [7], 是], [4, [4], [4], 是], [5, [5], [1], 是], [6, [6], [6], 是], [7, [7], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], 是], [1, [1], [5], 是], [2, [2], [2], 是], [3, [3], [7], 是], [4, [4], [4], 是], [5, [5], [1], 是], [6, [6], [6], 是], [7, [7], [3], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0049/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 50. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-well-definedness；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,7,6,5,4,3,2,1]}。請依 [kernel, cosets, representative_image_rows, well_defined, certificate] 欄序回答。

查看答案與解釋([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], 是], [1, [1], [7], 是], [2, [2], [6], 是], [3, [3], [5], 是], [4, [4], [4], 是], [5, [5], [3], 是], [6, [6], [2], 是], [7, [7], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)

Wave6第九層核對=([0], [[0], [1], [2], [3], [4], [5], [6], [7]], [[0, [0], [0], 是], [1, [1], [7], 是], [2, [2], [6], 是], [3, [3], [5], 是], [4, [4], [4], 是], [5, [5], [3], 是], [6, [6], [2], 是], [7, [7], [1], 是]], 是, bar_f_of_g_kernel_equals_f_of_g_for_every_representative)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0050/)

有限群第一同構定理 · qf-mod46-iso1-induced-well-definedness · candidate





## 51. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[0, 2], [1, 3]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, 是], [0, 1, 1, 1, 是], [1, 0, 1, 1, 是], [1, 1, 0, 0, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[0, 2], [1, 3]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, 是], [0, 1, 1, 1, 是], [1, 0, 1, 1, 是], [1, 1, 0, 0, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0051/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 52. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,1,2,3]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 0, 0, 是], [2, 3, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 0, 0, 是], [3, 2, 1, 1, 是], [3, 3, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 0, 0, 是], [2, 3, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 0, 0, 是], [3, 2, 1, 1, 是], [3, 3, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0052/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 53. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,3,2,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 3, 3, 是], [0, 2, 2, 2, 是], [0, 3, 1, 1, 是], [1, 0, 3, 3, 是], [1, 1, 2, 2, 是], [1, 2, 1, 1, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是], [2, 3, 3, 3, 是], [3, 0, 1, 1, 是], [3, 1, 0, 0, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 3, 3, 是], [0, 2, 2, 2, 是], [0, 3, 1, 1, 是], [1, 0, 3, 3, 是], [1, 1, 2, 2, 是], [1, 2, 1, 1, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是], [2, 3, 3, 3, 是], [3, 0, 1, 1, 是], [3, 1, 0, 0, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0053/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 54. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,1,2,3,4]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [0, 4, 4, 4, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 4, 4, 是], [1, 4, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 4, 4, 是], [2, 3, 0, 0, 是], [2, 4, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 4, 4, 是], [3, 2, 0, 0, 是], [3, 3, 1, 1, 是], [3, 4, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 0, 0, 是], [4, 2, 1, 1, 是], [4, 3, 2, 2, 是], [4, 4, 3, 3, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [0, 4, 4, 4, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 4, 4, 是], [1, 4, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 4, 4, 是], [2, 3, 0, 0, 是], [2, 4, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 4, 4, 是], [3, 2, 0, 0, 是], [3, 3, 1, 1, 是], [3, 4, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 0, 0, 是], [4, 2, 1, 1, 是], [4, 3, 2, 2, 是], [4, 4, 3, 3, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0054/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 55. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,2,4,1,3]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 1, 1, 是], [0, 4, 3, 3, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 1, 1, 是], [1, 3, 3, 3, 是], [1, 4, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 1, 1, 是], [2, 2, 3, 3, 是], [2, 3, 0, 0, 是], [2, 4, 2, 2, 是], [3, 0, 1, 1, 是], [3, 1, 3, 3, 是], [3, 2, 0, 0, 是], [3, 3, 2, 2, 是], [3, 4, 4, 4, 是], [4, 0, 3, 3, 是], [4, 1, 0, 0, 是], [4, 2, 2, 2, 是], [4, 3, 4, 4, 是], [4, 4, 1, 1, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 1, 1, 是], [0, 4, 3, 3, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 1, 1, 是], [1, 3, 3, 3, 是], [1, 4, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 1, 1, 是], [2, 2, 3, 3, 是], [2, 3, 0, 0, 是], [2, 4, 2, 2, 是], [3, 0, 1, 1, 是], [3, 1, 3, 3, 是], [3, 2, 0, 0, 是], [3, 3, 2, 2, 是], [3, 4, 4, 4, 是], [4, 0, 3, 3, 是], [4, 1, 0, 0, 是], [4, 2, 2, 2, 是], [4, 3, 4, 4, 是], [4, 4, 1, 1, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0055/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 56. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,3,1,4,2]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 3, 3, 是], [0, 2, 1, 1, 是], [0, 3, 4, 4, 是], [0, 4, 2, 2, 是], [1, 0, 3, 3, 是], [1, 1, 1, 1, 是], [1, 2, 4, 4, 是], [1, 3, 2, 2, 是], [1, 4, 0, 0, 是], [2, 0, 1, 1, 是], [2, 1, 4, 4, 是], [2, 2, 2, 2, 是], [2, 3, 0, 0, 是], [2, 4, 3, 3, 是], [3, 0, 4, 4, 是], [3, 1, 2, 2, 是], [3, 2, 0, 0, 是], [3, 3, 3, 3, 是], [3, 4, 1, 1, 是], [4, 0, 2, 2, 是], [4, 1, 0, 0, 是], [4, 2, 3, 3, 是], [4, 3, 1, 1, 是], [4, 4, 4, 4, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 3, 3, 是], [0, 2, 1, 1, 是], [0, 3, 4, 4, 是], [0, 4, 2, 2, 是], [1, 0, 3, 3, 是], [1, 1, 1, 1, 是], [1, 2, 4, 4, 是], [1, 3, 2, 2, 是], [1, 4, 0, 0, 是], [2, 0, 1, 1, 是], [2, 1, 4, 4, 是], [2, 2, 2, 2, 是], [2, 3, 0, 0, 是], [2, 4, 3, 3, 是], [3, 0, 4, 4, 是], [3, 1, 2, 2, 是], [3, 2, 0, 0, 是], [3, 3, 3, 3, 是], [3, 4, 1, 1, 是], [4, 0, 2, 2, 是], [4, 1, 0, 0, 是], [4, 2, 3, 3, 是], [4, 3, 1, 1, 是], [4, 4, 4, 4, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0056/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 57. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"target_identity":0,"mapping":[0,4,3,2,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 4, 4, 是], [0, 2, 3, 3, 是], [0, 3, 2, 2, 是], [0, 4, 1, 1, 是], [1, 0, 4, 4, 是], [1, 1, 3, 3, 是], [1, 2, 2, 2, 是], [1, 3, 1, 1, 是], [1, 4, 0, 0, 是], [2, 0, 3, 3, 是], [2, 1, 2, 2, 是], [2, 2, 1, 1, 是], [2, 3, 0, 0, 是], [2, 4, 4, 4, 是], [3, 0, 2, 2, 是], [3, 1, 1, 1, 是], [3, 2, 0, 0, 是], [3, 3, 4, 4, 是], [3, 4, 3, 3, 是], [4, 0, 1, 1, 是], [4, 1, 0, 0, 是], [4, 2, 4, 4, 是], [4, 3, 3, 3, 是], [4, 4, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 4, 4, 是], [0, 2, 3, 3, 是], [0, 3, 2, 2, 是], [0, 4, 1, 1, 是], [1, 0, 4, 4, 是], [1, 1, 3, 3, 是], [1, 2, 2, 2, 是], [1, 3, 1, 1, 是], [1, 4, 0, 0, 是], [2, 0, 3, 3, 是], [2, 1, 2, 2, 是], [2, 2, 1, 1, 是], [2, 3, 0, 0, 是], [2, 4, 4, 4, 是], [3, 0, 2, 2, 是], [3, 1, 1, 1, 是], [3, 2, 0, 0, 是], [3, 3, 4, 4, 是], [3, 4, 3, 3, 是], [4, 0, 1, 1, 是], [4, 1, 0, 0, 是], [4, 2, 4, 4, 是], [4, 3, 3, 3, 是], [4, 4, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0057/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 58. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[0, 2, 4], [1, 3, 5]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, 是], [0, 1, 1, 1, 是], [1, 0, 1, 1, 是], [1, 1, 0, 0, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[0, 2, 4], [1, 3, 5]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, 是], [0, 1, 1, 1, 是], [1, 0, 1, 1, 是], [1, 1, 0, 0, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0058/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 59. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"target_identity":0,"mapping":[0,1,2,0,1,2]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 0, 0, 是], [2, 2, 1, 1, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 0, 0, 是], [2, 2, 1, 1, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0059/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 60. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2],[1,2,0],[2,0,1]],"target_identity":0,"mapping":[0,2,1,0,2,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 2, 2, 是], [0, 2, 1, 1, 是], [1, 0, 2, 2, 是], [1, 1, 1, 1, 是], [1, 2, 0, 0, 是], [2, 0, 1, 1, 是], [2, 1, 0, 0, 是], [2, 2, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 2, 2, 是], [0, 2, 1, 1, 是], [1, 0, 2, 2, 是], [1, 1, 1, 1, 是], [1, 2, 0, 0, 是], [2, 0, 1, 1, 是], [2, 1, 0, 0, 是], [2, 2, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0060/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 61. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,1,2,3,4,5]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [0, 4, 4, 4, 是], [0, 5, 5, 5, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 4, 4, 是], [1, 4, 5, 5, 是], [1, 5, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 4, 4, 是], [2, 3, 5, 5, 是], [2, 4, 0, 0, 是], [2, 5, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 4, 4, 是], [3, 2, 5, 5, 是], [3, 3, 0, 0, 是], [3, 4, 1, 1, 是], [3, 5, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 5, 5, 是], [4, 2, 0, 0, 是], [4, 3, 1, 1, 是], [4, 4, 2, 2, 是], [4, 5, 3, 3, 是], [5, 0, 5, 5, 是], [5, 1, 0, 0, 是], [5, 2, 1, 1, 是], [5, 3, 2, 2, 是], [5, 4, 3, 3, 是], [5, 5, 4, 4, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [0, 4, 4, 4, 是], [0, 5, 5, 5, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 4, 4, 是], [1, 4, 5, 5, 是], [1, 5, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 4, 4, 是], [2, 3, 5, 5, 是], [2, 4, 0, 0, 是], [2, 5, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 4, 4, 是], [3, 2, 5, 5, 是], [3, 3, 0, 0, 是], [3, 4, 1, 1, 是], [3, 5, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 5, 5, 是], [4, 2, 0, 0, 是], [4, 3, 1, 1, 是], [4, 4, 2, 2, 是], [4, 5, 3, 3, 是], [5, 0, 5, 5, 是], [5, 1, 0, 0, 是], [5, 2, 1, 1, 是], [5, 3, 2, 2, 是], [5, 4, 3, 3, 是], [5, 5, 4, 4, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0061/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 62. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,5,4,3,2,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 5, 5, 是], [0, 2, 4, 4, 是], [0, 3, 3, 3, 是], [0, 4, 2, 2, 是], [0, 5, 1, 1, 是], [1, 0, 5, 5, 是], [1, 1, 4, 4, 是], [1, 2, 3, 3, 是], [1, 3, 2, 2, 是], [1, 4, 1, 1, 是], [1, 5, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 3, 3, 是], [2, 2, 2, 2, 是], [2, 3, 1, 1, 是], [2, 4, 0, 0, 是], [2, 5, 5, 5, 是], [3, 0, 3, 3, 是], [3, 1, 2, 2, 是], [3, 2, 1, 1, 是], [3, 3, 0, 0, 是], [3, 4, 5, 5, 是], [3, 5, 4, 4, 是], [4, 0, 2, 2, 是], [4, 1, 1, 1, 是], [4, 2, 0, 0, 是], [4, 3, 5, 5, 是], [4, 4, 4, 4, 是], [4, 5, 3, 3, 是], [5, 0, 1, 1, 是], [5, 1, 0, 0, 是], [5, 2, 5, 5, 是], [5, 3, 4, 4, 是], [5, 4, 3, 3, 是], [5, 5, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 5, 5, 是], [0, 2, 4, 4, 是], [0, 3, 3, 3, 是], [0, 4, 2, 2, 是], [0, 5, 1, 1, 是], [1, 0, 5, 5, 是], [1, 1, 4, 4, 是], [1, 2, 3, 3, 是], [1, 3, 2, 2, 是], [1, 4, 1, 1, 是], [1, 5, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 3, 3, 是], [2, 2, 2, 2, 是], [2, 3, 1, 1, 是], [2, 4, 0, 0, 是], [2, 5, 5, 5, 是], [3, 0, 3, 3, 是], [3, 1, 2, 2, 是], [3, 2, 1, 1, 是], [3, 3, 0, 0, 是], [3, 4, 5, 5, 是], [3, 5, 4, 4, 是], [4, 0, 2, 2, 是], [4, 1, 1, 1, 是], [4, 2, 0, 0, 是], [4, 3, 5, 5, 是], [4, 4, 4, 4, 是], [4, 5, 3, 3, 是], [5, 0, 1, 1, 是], [5, 1, 0, 0, 是], [5, 2, 5, 5, 是], [5, 3, 4, 4, 是], [5, 4, 3, 3, 是], [5, 5, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0062/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 63. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,1,2,3,4,5,6]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [0, 4, 4, 4, 是], [0, 5, 5, 5, 是], [0, 6, 6, 6, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 4, 4, 是], [1, 4, 5, 5, 是], [1, 5, 6, 6, 是], [1, 6, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 4, 4, 是], [2, 3, 5, 5, 是], [2, 4, 6, 6, 是], [2, 5, 0, 0, 是], [2, 6, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 4, 4, 是], [3, 2, 5, 5, 是], [3, 3, 6, 6, 是], [3, 4, 0, 0, 是], [3, 5, 1, 1, 是], [3, 6, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 5, 5, 是], [4, 2, 6, 6, 是], [4, 3, 0, 0, 是], [4, 4, 1, 1, 是], [4, 5, 2, 2, 是], [4, 6, 3, 3, 是], [5, 0, 5, 5, 是], [5, 1, 6, 6, 是], [5, 2, 0, 0, 是], [5, 3, 1, 1, 是], [5, 4, 2, 2, 是], [5, 5, 3, 3, 是], [5, 6, 4, 4, 是], [6, 0, 6, 6, 是], [6, 1, 0, 0, 是], [6, 2, 1, 1, 是], [6, 3, 2, 2, 是], [6, 4, 3, 3, 是], [6, 5, 4, 4, 是], [6, 6, 5, 5, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [0, 4, 4, 4, 是], [0, 5, 5, 5, 是], [0, 6, 6, 6, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 4, 4, 是], [1, 4, 5, 5, 是], [1, 5, 6, 6, 是], [1, 6, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 4, 4, 是], [2, 3, 5, 5, 是], [2, 4, 6, 6, 是], [2, 5, 0, 0, 是], [2, 6, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 4, 4, 是], [3, 2, 5, 5, 是], [3, 3, 6, 6, 是], [3, 4, 0, 0, 是], [3, 5, 1, 1, 是], [3, 6, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 5, 5, 是], [4, 2, 6, 6, 是], [4, 3, 0, 0, 是], [4, 4, 1, 1, 是], [4, 5, 2, 2, 是], [4, 6, 3, 3, 是], [5, 0, 5, 5, 是], [5, 1, 6, 6, 是], [5, 2, 0, 0, 是], [5, 3, 1, 1, 是], [5, 4, 2, 2, 是], [5, 5, 3, 3, 是], [5, 6, 4, 4, 是], [6, 0, 6, 6, 是], [6, 1, 0, 0, 是], [6, 2, 1, 1, 是], [6, 3, 2, 2, 是], [6, 4, 3, 3, 是], [6, 5, 4, 4, 是], [6, 6, 5, 5, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0063/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 64. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,2,4,6,1,3,5]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 6, 6, 是], [0, 4, 1, 1, 是], [0, 5, 3, 3, 是], [0, 6, 5, 5, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 6, 6, 是], [1, 3, 1, 1, 是], [1, 4, 3, 3, 是], [1, 5, 5, 5, 是], [1, 6, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 6, 6, 是], [2, 2, 1, 1, 是], [2, 3, 3, 3, 是], [2, 4, 5, 5, 是], [2, 5, 0, 0, 是], [2, 6, 2, 2, 是], [3, 0, 6, 6, 是], [3, 1, 1, 1, 是], [3, 2, 3, 3, 是], [3, 3, 5, 5, 是], [3, 4, 0, 0, 是], [3, 5, 2, 2, 是], [3, 6, 4, 4, 是], [4, 0, 1, 1, 是], [4, 1, 3, 3, 是], [4, 2, 5, 5, 是], [4, 3, 0, 0, 是], [4, 4, 2, 2, 是], [4, 5, 4, 4, 是], [4, 6, 6, 6, 是], [5, 0, 3, 3, 是], [5, 1, 5, 5, 是], [5, 2, 0, 0, 是], [5, 3, 2, 2, 是], [5, 4, 4, 4, 是], [5, 5, 6, 6, 是], [5, 6, 1, 1, 是], [6, 0, 5, 5, 是], [6, 1, 0, 0, 是], [6, 2, 2, 2, 是], [6, 3, 4, 4, 是], [6, 4, 6, 6, 是], [6, 5, 1, 1, 是], [6, 6, 3, 3, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 6, 6, 是], [0, 4, 1, 1, 是], [0, 5, 3, 3, 是], [0, 6, 5, 5, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 6, 6, 是], [1, 3, 1, 1, 是], [1, 4, 3, 3, 是], [1, 5, 5, 5, 是], [1, 6, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 6, 6, 是], [2, 2, 1, 1, 是], [2, 3, 3, 3, 是], [2, 4, 5, 5, 是], [2, 5, 0, 0, 是], [2, 6, 2, 2, 是], [3, 0, 6, 6, 是], [3, 1, 1, 1, 是], [3, 2, 3, 3, 是], [3, 3, 5, 5, 是], [3, 4, 0, 0, 是], [3, 5, 2, 2, 是], [3, 6, 4, 4, 是], [4, 0, 1, 1, 是], [4, 1, 3, 3, 是], [4, 2, 5, 5, 是], [4, 3, 0, 0, 是], [4, 4, 2, 2, 是], [4, 5, 4, 4, 是], [4, 6, 6, 6, 是], [5, 0, 3, 3, 是], [5, 1, 5, 5, 是], [5, 2, 0, 0, 是], [5, 3, 2, 2, 是], [5, 4, 4, 4, 是], [5, 5, 6, 6, 是], [5, 6, 1, 1, 是], [6, 0, 5, 5, 是], [6, 1, 0, 0, 是], [6, 2, 2, 2, 是], [6, 3, 4, 4, 是], [6, 4, 6, 6, 是], [6, 5, 1, 1, 是], [6, 6, 3, 3, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0064/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 65. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,3,6,2,5,1,4]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 3, 3, 是], [0, 2, 6, 6, 是], [0, 3, 2, 2, 是], [0, 4, 5, 5, 是], [0, 5, 1, 1, 是], [0, 6, 4, 4, 是], [1, 0, 3, 3, 是], [1, 1, 6, 6, 是], [1, 2, 2, 2, 是], [1, 3, 5, 5, 是], [1, 4, 1, 1, 是], [1, 5, 4, 4, 是], [1, 6, 0, 0, 是], [2, 0, 6, 6, 是], [2, 1, 2, 2, 是], [2, 2, 5, 5, 是], [2, 3, 1, 1, 是], [2, 4, 4, 4, 是], [2, 5, 0, 0, 是], [2, 6, 3, 3, 是], [3, 0, 2, 2, 是], [3, 1, 5, 5, 是], [3, 2, 1, 1, 是], [3, 3, 4, 4, 是], [3, 4, 0, 0, 是], [3, 5, 3, 3, 是], [3, 6, 6, 6, 是], [4, 0, 5, 5, 是], [4, 1, 1, 1, 是], [4, 2, 4, 4, 是], [4, 3, 0, 0, 是], [4, 4, 3, 3, 是], [4, 5, 6, 6, 是], [4, 6, 2, 2, 是], [5, 0, 1, 1, 是], [5, 1, 4, 4, 是], [5, 2, 0, 0, 是], [5, 3, 3, 3, 是], [5, 4, 6, 6, 是], [5, 5, 2, 2, 是], [5, 6, 5, 5, 是], [6, 0, 4, 4, 是], [6, 1, 0, 0, 是], [6, 2, 3, 3, 是], [6, 3, 6, 6, 是], [6, 4, 2, 2, 是], [6, 5, 5, 5, 是], [6, 6, 1, 1, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 3, 3, 是], [0, 2, 6, 6, 是], [0, 3, 2, 2, 是], [0, 4, 5, 5, 是], [0, 5, 1, 1, 是], [0, 6, 4, 4, 是], [1, 0, 3, 3, 是], [1, 1, 6, 6, 是], [1, 2, 2, 2, 是], [1, 3, 5, 5, 是], [1, 4, 1, 1, 是], [1, 5, 4, 4, 是], [1, 6, 0, 0, 是], [2, 0, 6, 6, 是], [2, 1, 2, 2, 是], [2, 2, 5, 5, 是], [2, 3, 1, 1, 是], [2, 4, 4, 4, 是], [2, 5, 0, 0, 是], [2, 6, 3, 3, 是], [3, 0, 2, 2, 是], [3, 1, 5, 5, 是], [3, 2, 1, 1, 是], [3, 3, 4, 4, 是], [3, 4, 0, 0, 是], [3, 5, 3, 3, 是], [3, 6, 6, 6, 是], [4, 0, 5, 5, 是], [4, 1, 1, 1, 是], [4, 2, 4, 4, 是], [4, 3, 0, 0, 是], [4, 4, 3, 3, 是], [4, 5, 6, 6, 是], [4, 6, 2, 2, 是], [5, 0, 1, 1, 是], [5, 1, 4, 4, 是], [5, 2, 0, 0, 是], [5, 3, 3, 3, 是], [5, 4, 6, 6, 是], [5, 5, 2, 2, 是], [5, 6, 5, 5, 是], [6, 0, 4, 4, 是], [6, 1, 0, 0, 是], [6, 2, 3, 3, 是], [6, 3, 6, 6, 是], [6, 4, 2, 2, 是], [6, 5, 5, 5, 是], [6, 6, 1, 1, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0065/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 66. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,4,1,5,2,6,3]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 4, 4, 是], [0, 2, 1, 1, 是], [0, 3, 5, 5, 是], [0, 4, 2, 2, 是], [0, 5, 6, 6, 是], [0, 6, 3, 3, 是], [1, 0, 4, 4, 是], [1, 1, 1, 1, 是], [1, 2, 5, 5, 是], [1, 3, 2, 2, 是], [1, 4, 6, 6, 是], [1, 5, 3, 3, 是], [1, 6, 0, 0, 是], [2, 0, 1, 1, 是], [2, 1, 5, 5, 是], [2, 2, 2, 2, 是], [2, 3, 6, 6, 是], [2, 4, 3, 3, 是], [2, 5, 0, 0, 是], [2, 6, 4, 4, 是], [3, 0, 5, 5, 是], [3, 1, 2, 2, 是], [3, 2, 6, 6, 是], [3, 3, 3, 3, 是], [3, 4, 0, 0, 是], [3, 5, 4, 4, 是], [3, 6, 1, 1, 是], [4, 0, 2, 2, 是], [4, 1, 6, 6, 是], [4, 2, 3, 3, 是], [4, 3, 0, 0, 是], [4, 4, 4, 4, 是], [4, 5, 1, 1, 是], [4, 6, 5, 5, 是], [5, 0, 6, 6, 是], [5, 1, 3, 3, 是], [5, 2, 0, 0, 是], [5, 3, 4, 4, 是], [5, 4, 1, 1, 是], [5, 5, 5, 5, 是], [5, 6, 2, 2, 是], [6, 0, 3, 3, 是], [6, 1, 0, 0, 是], [6, 2, 4, 4, 是], [6, 3, 1, 1, 是], [6, 4, 5, 5, 是], [6, 5, 2, 2, 是], [6, 6, 6, 6, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 4, 4, 是], [0, 2, 1, 1, 是], [0, 3, 5, 5, 是], [0, 4, 2, 2, 是], [0, 5, 6, 6, 是], [0, 6, 3, 3, 是], [1, 0, 4, 4, 是], [1, 1, 1, 1, 是], [1, 2, 5, 5, 是], [1, 3, 2, 2, 是], [1, 4, 6, 6, 是], [1, 5, 3, 3, 是], [1, 6, 0, 0, 是], [2, 0, 1, 1, 是], [2, 1, 5, 5, 是], [2, 2, 2, 2, 是], [2, 3, 6, 6, 是], [2, 4, 3, 3, 是], [2, 5, 0, 0, 是], [2, 6, 4, 4, 是], [3, 0, 5, 5, 是], [3, 1, 2, 2, 是], [3, 2, 6, 6, 是], [3, 3, 3, 3, 是], [3, 4, 0, 0, 是], [3, 5, 4, 4, 是], [3, 6, 1, 1, 是], [4, 0, 2, 2, 是], [4, 1, 6, 6, 是], [4, 2, 3, 3, 是], [4, 3, 0, 0, 是], [4, 4, 4, 4, 是], [4, 5, 1, 1, 是], [4, 6, 5, 5, 是], [5, 0, 6, 6, 是], [5, 1, 3, 3, 是], [5, 2, 0, 0, 是], [5, 3, 4, 4, 是], [5, 4, 1, 1, 是], [5, 5, 5, 5, 是], [5, 6, 2, 2, 是], [6, 0, 3, 3, 是], [6, 1, 0, 0, 是], [6, 2, 4, 4, 是], [6, 3, 1, 1, 是], [6, 4, 5, 5, 是], [6, 5, 2, 2, 是], [6, 6, 6, 6, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0066/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 67. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,5,3,1,6,4,2]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 5, 5, 是], [0, 2, 3, 3, 是], [0, 3, 1, 1, 是], [0, 4, 6, 6, 是], [0, 5, 4, 4, 是], [0, 6, 2, 2, 是], [1, 0, 5, 5, 是], [1, 1, 3, 3, 是], [1, 2, 1, 1, 是], [1, 3, 6, 6, 是], [1, 4, 4, 4, 是], [1, 5, 2, 2, 是], [1, 6, 0, 0, 是], [2, 0, 3, 3, 是], [2, 1, 1, 1, 是], [2, 2, 6, 6, 是], [2, 3, 4, 4, 是], [2, 4, 2, 2, 是], [2, 5, 0, 0, 是], [2, 6, 5, 5, 是], [3, 0, 1, 1, 是], [3, 1, 6, 6, 是], [3, 2, 4, 4, 是], [3, 3, 2, 2, 是], [3, 4, 0, 0, 是], [3, 5, 5, 5, 是], [3, 6, 3, 3, 是], [4, 0, 6, 6, 是], [4, 1, 4, 4, 是], [4, 2, 2, 2, 是], [4, 3, 0, 0, 是], [4, 4, 5, 5, 是], [4, 5, 3, 3, 是], [4, 6, 1, 1, 是], [5, 0, 4, 4, 是], [5, 1, 2, 2, 是], [5, 2, 0, 0, 是], [5, 3, 5, 5, 是], [5, 4, 3, 3, 是], [5, 5, 1, 1, 是], [5, 6, 6, 6, 是], [6, 0, 2, 2, 是], [6, 1, 0, 0, 是], [6, 2, 5, 5, 是], [6, 3, 3, 3, 是], [6, 4, 1, 1, 是], [6, 5, 6, 6, 是], [6, 6, 4, 4, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 5, 5, 是], [0, 2, 3, 3, 是], [0, 3, 1, 1, 是], [0, 4, 6, 6, 是], [0, 5, 4, 4, 是], [0, 6, 2, 2, 是], [1, 0, 5, 5, 是], [1, 1, 3, 3, 是], [1, 2, 1, 1, 是], [1, 3, 6, 6, 是], [1, 4, 4, 4, 是], [1, 5, 2, 2, 是], [1, 6, 0, 0, 是], [2, 0, 3, 3, 是], [2, 1, 1, 1, 是], [2, 2, 6, 6, 是], [2, 3, 4, 4, 是], [2, 4, 2, 2, 是], [2, 5, 0, 0, 是], [2, 6, 5, 5, 是], [3, 0, 1, 1, 是], [3, 1, 6, 6, 是], [3, 2, 4, 4, 是], [3, 3, 2, 2, 是], [3, 4, 0, 0, 是], [3, 5, 5, 5, 是], [3, 6, 3, 3, 是], [4, 0, 6, 6, 是], [4, 1, 4, 4, 是], [4, 2, 2, 2, 是], [4, 3, 0, 0, 是], [4, 4, 5, 5, 是], [4, 5, 3, 3, 是], [4, 6, 1, 1, 是], [5, 0, 4, 4, 是], [5, 1, 2, 2, 是], [5, 2, 0, 0, 是], [5, 3, 5, 5, 是], [5, 4, 3, 3, 是], [5, 5, 1, 1, 是], [5, 6, 6, 6, 是], [6, 0, 2, 2, 是], [6, 1, 0, 0, 是], [6, 2, 5, 5, 是], [6, 3, 3, 3, 是], [6, 4, 1, 1, 是], [6, 5, 6, 6, 是], [6, 6, 4, 4, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0067/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 68. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"target_identity":0,"mapping":[0,6,5,4,3,2,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 6, 6, 是], [0, 2, 5, 5, 是], [0, 3, 4, 4, 是], [0, 4, 3, 3, 是], [0, 5, 2, 2, 是], [0, 6, 1, 1, 是], [1, 0, 6, 6, 是], [1, 1, 5, 5, 是], [1, 2, 4, 4, 是], [1, 3, 3, 3, 是], [1, 4, 2, 2, 是], [1, 5, 1, 1, 是], [1, 6, 0, 0, 是], [2, 0, 5, 5, 是], [2, 1, 4, 4, 是], [2, 2, 3, 3, 是], [2, 3, 2, 2, 是], [2, 4, 1, 1, 是], [2, 5, 0, 0, 是], [2, 6, 6, 6, 是], [3, 0, 4, 4, 是], [3, 1, 3, 3, 是], [3, 2, 2, 2, 是], [3, 3, 1, 1, 是], [3, 4, 0, 0, 是], [3, 5, 6, 6, 是], [3, 6, 5, 5, 是], [4, 0, 3, 3, 是], [4, 1, 2, 2, 是], [4, 2, 1, 1, 是], [4, 3, 0, 0, 是], [4, 4, 6, 6, 是], [4, 5, 5, 5, 是], [4, 6, 4, 4, 是], [5, 0, 2, 2, 是], [5, 1, 1, 1, 是], [5, 2, 0, 0, 是], [5, 3, 6, 6, 是], [5, 4, 5, 5, 是], [5, 5, 4, 4, 是], [5, 6, 3, 3, 是], [6, 0, 1, 1, 是], [6, 1, 0, 0, 是], [6, 2, 6, 6, 是], [6, 3, 5, 5, 是], [6, 4, 4, 4, 是], [6, 5, 3, 3, 是], [6, 6, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 6, 6, 是], [0, 2, 5, 5, 是], [0, 3, 4, 4, 是], [0, 4, 3, 3, 是], [0, 5, 2, 2, 是], [0, 6, 1, 1, 是], [1, 0, 6, 6, 是], [1, 1, 5, 5, 是], [1, 2, 4, 4, 是], [1, 3, 3, 3, 是], [1, 4, 2, 2, 是], [1, 5, 1, 1, 是], [1, 6, 0, 0, 是], [2, 0, 5, 5, 是], [2, 1, 4, 4, 是], [2, 2, 3, 3, 是], [2, 3, 2, 2, 是], [2, 4, 1, 1, 是], [2, 5, 0, 0, 是], [2, 6, 6, 6, 是], [3, 0, 4, 4, 是], [3, 1, 3, 3, 是], [3, 2, 2, 2, 是], [3, 3, 1, 1, 是], [3, 4, 0, 0, 是], [3, 5, 6, 6, 是], [3, 6, 5, 5, 是], [4, 0, 3, 3, 是], [4, 1, 2, 2, 是], [4, 2, 1, 1, 是], [4, 3, 0, 0, 是], [4, 4, 6, 6, 是], [4, 5, 5, 5, 是], [4, 6, 4, 4, 是], [5, 0, 2, 2, 是], [5, 1, 1, 1, 是], [5, 2, 0, 0, 是], [5, 3, 6, 6, 是], [5, 4, 5, 5, 是], [5, 5, 4, 4, 是], [5, 6, 3, 3, 是], [6, 0, 1, 1, 是], [6, 1, 0, 0, 是], [6, 2, 6, 6, 是], [6, 3, 5, 5, 是], [6, 4, 4, 4, 是], [6, 5, 3, 3, 是], [6, 6, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0068/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 69. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1],[1,0]],"target_identity":0,"mapping":[0,1,0,1,0,1,0,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[0, 2, 4, 6], [1, 3, 5, 7]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, 是], [0, 1, 1, 1, 是], [1, 0, 1, 1, 是], [1, 1, 0, 0, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[0, 2, 4, 6], [1, 3, 5, 7]], [0, 1], [[0, 1], [1, 0]], [0, 1], [[0, 0, 0, 0, 是], [0, 1, 1, 1, 是], [1, 0, 1, 1, 是], [1, 1, 0, 0, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0069/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 70. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,1,2,3,0,1,2,3]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 0, 0, 是], [2, 3, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 0, 0, 是], [3, 2, 1, 1, 是], [3, 3, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 0, 0, 是], [2, 3, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 0, 0, 是], [3, 2, 1, 1, 是], [3, 3, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0070/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 71. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,3,2,1,0,3,2,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 3, 3, 是], [0, 2, 2, 2, 是], [0, 3, 1, 1, 是], [1, 0, 3, 3, 是], [1, 1, 2, 2, 是], [1, 2, 1, 1, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是], [2, 3, 3, 3, 是], [3, 0, 1, 1, 是], [3, 1, 0, 0, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 3, 3, 是], [0, 2, 2, 2, 是], [0, 3, 1, 1, 是], [1, 0, 3, 3, 是], [1, 1, 2, 2, 是], [1, 2, 1, 1, 是], [1, 3, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 1, 1, 是], [2, 2, 0, 0, 是], [2, 3, 3, 3, 是], [3, 0, 1, 1, 是], [3, 1, 0, 0, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0071/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 72. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,1,2,3,4,5,6,7]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [0, 4, 4, 4, 是], [0, 5, 5, 5, 是], [0, 6, 6, 6, 是], [0, 7, 7, 7, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 4, 4, 是], [1, 4, 5, 5, 是], [1, 5, 6, 6, 是], [1, 6, 7, 7, 是], [1, 7, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 4, 4, 是], [2, 3, 5, 5, 是], [2, 4, 6, 6, 是], [2, 5, 7, 7, 是], [2, 6, 0, 0, 是], [2, 7, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 4, 4, 是], [3, 2, 5, 5, 是], [3, 3, 6, 6, 是], [3, 4, 7, 7, 是], [3, 5, 0, 0, 是], [3, 6, 1, 1, 是], [3, 7, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 5, 5, 是], [4, 2, 6, 6, 是], [4, 3, 7, 7, 是], [4, 4, 0, 0, 是], [4, 5, 1, 1, 是], [4, 6, 2, 2, 是], [4, 7, 3, 3, 是], [5, 0, 5, 5, 是], [5, 1, 6, 6, 是], [5, 2, 7, 7, 是], [5, 3, 0, 0, 是], [5, 4, 1, 1, 是], [5, 5, 2, 2, 是], [5, 6, 3, 3, 是], [5, 7, 4, 4, 是], [6, 0, 6, 6, 是], [6, 1, 7, 7, 是], [6, 2, 0, 0, 是], [6, 3, 1, 1, 是], [6, 4, 2, 2, 是], [6, 5, 3, 3, 是], [6, 6, 4, 4, 是], [6, 7, 5, 5, 是], [7, 0, 7, 7, 是], [7, 1, 0, 0, 是], [7, 2, 1, 1, 是], [7, 3, 2, 2, 是], [7, 4, 3, 3, 是], [7, 5, 4, 4, 是], [7, 6, 5, 5, 是], [7, 7, 6, 6, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 1, 1, 是], [0, 2, 2, 2, 是], [0, 3, 3, 3, 是], [0, 4, 4, 4, 是], [0, 5, 5, 5, 是], [0, 6, 6, 6, 是], [0, 7, 7, 7, 是], [1, 0, 1, 1, 是], [1, 1, 2, 2, 是], [1, 2, 3, 3, 是], [1, 3, 4, 4, 是], [1, 4, 5, 5, 是], [1, 5, 6, 6, 是], [1, 6, 7, 7, 是], [1, 7, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 3, 3, 是], [2, 2, 4, 4, 是], [2, 3, 5, 5, 是], [2, 4, 6, 6, 是], [2, 5, 7, 7, 是], [2, 6, 0, 0, 是], [2, 7, 1, 1, 是], [3, 0, 3, 3, 是], [3, 1, 4, 4, 是], [3, 2, 5, 5, 是], [3, 3, 6, 6, 是], [3, 4, 7, 7, 是], [3, 5, 0, 0, 是], [3, 6, 1, 1, 是], [3, 7, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 5, 5, 是], [4, 2, 6, 6, 是], [4, 3, 7, 7, 是], [4, 4, 0, 0, 是], [4, 5, 1, 1, 是], [4, 6, 2, 2, 是], [4, 7, 3, 3, 是], [5, 0, 5, 5, 是], [5, 1, 6, 6, 是], [5, 2, 7, 7, 是], [5, 3, 0, 0, 是], [5, 4, 1, 1, 是], [5, 5, 2, 2, 是], [5, 6, 3, 3, 是], [5, 7, 4, 4, 是], [6, 0, 6, 6, 是], [6, 1, 7, 7, 是], [6, 2, 0, 0, 是], [6, 3, 1, 1, 是], [6, 4, 2, 2, 是], [6, 5, 3, 3, 是], [6, 6, 4, 4, 是], [6, 7, 5, 5, 是], [7, 0, 7, 7, 是], [7, 1, 0, 0, 是], [7, 2, 1, 1, 是], [7, 3, 2, 2, 是], [7, 4, 3, 3, 是], [7, 5, 4, 4, 是], [7, 6, 5, 5, 是], [7, 7, 6, 6, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0072/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 73. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,3,6,1,4,7,2,5]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 3, 3, 是], [0, 2, 6, 6, 是], [0, 3, 1, 1, 是], [0, 4, 4, 4, 是], [0, 5, 7, 7, 是], [0, 6, 2, 2, 是], [0, 7, 5, 5, 是], [1, 0, 3, 3, 是], [1, 1, 6, 6, 是], [1, 2, 1, 1, 是], [1, 3, 4, 4, 是], [1, 4, 7, 7, 是], [1, 5, 2, 2, 是], [1, 6, 5, 5, 是], [1, 7, 0, 0, 是], [2, 0, 6, 6, 是], [2, 1, 1, 1, 是], [2, 2, 4, 4, 是], [2, 3, 7, 7, 是], [2, 4, 2, 2, 是], [2, 5, 5, 5, 是], [2, 6, 0, 0, 是], [2, 7, 3, 3, 是], [3, 0, 1, 1, 是], [3, 1, 4, 4, 是], [3, 2, 7, 7, 是], [3, 3, 2, 2, 是], [3, 4, 5, 5, 是], [3, 5, 0, 0, 是], [3, 6, 3, 3, 是], [3, 7, 6, 6, 是], [4, 0, 4, 4, 是], [4, 1, 7, 7, 是], [4, 2, 2, 2, 是], [4, 3, 5, 5, 是], [4, 4, 0, 0, 是], [4, 5, 3, 3, 是], [4, 6, 6, 6, 是], [4, 7, 1, 1, 是], [5, 0, 7, 7, 是], [5, 1, 2, 2, 是], [5, 2, 5, 5, 是], [5, 3, 0, 0, 是], [5, 4, 3, 3, 是], [5, 5, 6, 6, 是], [5, 6, 1, 1, 是], [5, 7, 4, 4, 是], [6, 0, 2, 2, 是], [6, 1, 5, 5, 是], [6, 2, 0, 0, 是], [6, 3, 3, 3, 是], [6, 4, 6, 6, 是], [6, 5, 1, 1, 是], [6, 6, 4, 4, 是], [6, 7, 7, 7, 是], [7, 0, 5, 5, 是], [7, 1, 0, 0, 是], [7, 2, 3, 3, 是], [7, 3, 6, 6, 是], [7, 4, 1, 1, 是], [7, 5, 4, 4, 是], [7, 6, 7, 7, 是], [7, 7, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 3, 3, 是], [0, 2, 6, 6, 是], [0, 3, 1, 1, 是], [0, 4, 4, 4, 是], [0, 5, 7, 7, 是], [0, 6, 2, 2, 是], [0, 7, 5, 5, 是], [1, 0, 3, 3, 是], [1, 1, 6, 6, 是], [1, 2, 1, 1, 是], [1, 3, 4, 4, 是], [1, 4, 7, 7, 是], [1, 5, 2, 2, 是], [1, 6, 5, 5, 是], [1, 7, 0, 0, 是], [2, 0, 6, 6, 是], [2, 1, 1, 1, 是], [2, 2, 4, 4, 是], [2, 3, 7, 7, 是], [2, 4, 2, 2, 是], [2, 5, 5, 5, 是], [2, 6, 0, 0, 是], [2, 7, 3, 3, 是], [3, 0, 1, 1, 是], [3, 1, 4, 4, 是], [3, 2, 7, 7, 是], [3, 3, 2, 2, 是], [3, 4, 5, 5, 是], [3, 5, 0, 0, 是], [3, 6, 3, 3, 是], [3, 7, 6, 6, 是], [4, 0, 4, 4, 是], [4, 1, 7, 7, 是], [4, 2, 2, 2, 是], [4, 3, 5, 5, 是], [4, 4, 0, 0, 是], [4, 5, 3, 3, 是], [4, 6, 6, 6, 是], [4, 7, 1, 1, 是], [5, 0, 7, 7, 是], [5, 1, 2, 2, 是], [5, 2, 5, 5, 是], [5, 3, 0, 0, 是], [5, 4, 3, 3, 是], [5, 5, 6, 6, 是], [5, 6, 1, 1, 是], [5, 7, 4, 4, 是], [6, 0, 2, 2, 是], [6, 1, 5, 5, 是], [6, 2, 0, 0, 是], [6, 3, 3, 3, 是], [6, 4, 6, 6, 是], [6, 5, 1, 1, 是], [6, 6, 4, 4, 是], [6, 7, 7, 7, 是], [7, 0, 5, 5, 是], [7, 1, 0, 0, 是], [7, 2, 3, 3, 是], [7, 3, 6, 6, 是], [7, 4, 1, 1, 是], [7, 5, 4, 4, 是], [7, 6, 7, 7, 是], [7, 7, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0073/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 74. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,5,2,7,4,1,6,3]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 5, 5, 是], [0, 2, 2, 2, 是], [0, 3, 7, 7, 是], [0, 4, 4, 4, 是], [0, 5, 1, 1, 是], [0, 6, 6, 6, 是], [0, 7, 3, 3, 是], [1, 0, 5, 5, 是], [1, 1, 2, 2, 是], [1, 2, 7, 7, 是], [1, 3, 4, 4, 是], [1, 4, 1, 1, 是], [1, 5, 6, 6, 是], [1, 6, 3, 3, 是], [1, 7, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 7, 7, 是], [2, 2, 4, 4, 是], [2, 3, 1, 1, 是], [2, 4, 6, 6, 是], [2, 5, 3, 3, 是], [2, 6, 0, 0, 是], [2, 7, 5, 5, 是], [3, 0, 7, 7, 是], [3, 1, 4, 4, 是], [3, 2, 1, 1, 是], [3, 3, 6, 6, 是], [3, 4, 3, 3, 是], [3, 5, 0, 0, 是], [3, 6, 5, 5, 是], [3, 7, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 1, 1, 是], [4, 2, 6, 6, 是], [4, 3, 3, 3, 是], [4, 4, 0, 0, 是], [4, 5, 5, 5, 是], [4, 6, 2, 2, 是], [4, 7, 7, 7, 是], [5, 0, 1, 1, 是], [5, 1, 6, 6, 是], [5, 2, 3, 3, 是], [5, 3, 0, 0, 是], [5, 4, 5, 5, 是], [5, 5, 2, 2, 是], [5, 6, 7, 7, 是], [5, 7, 4, 4, 是], [6, 0, 6, 6, 是], [6, 1, 3, 3, 是], [6, 2, 0, 0, 是], [6, 3, 5, 5, 是], [6, 4, 2, 2, 是], [6, 5, 7, 7, 是], [6, 6, 4, 4, 是], [6, 7, 1, 1, 是], [7, 0, 3, 3, 是], [7, 1, 0, 0, 是], [7, 2, 5, 5, 是], [7, 3, 2, 2, 是], [7, 4, 7, 7, 是], [7, 5, 4, 4, 是], [7, 6, 1, 1, 是], [7, 7, 6, 6, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 5, 5, 是], [0, 2, 2, 2, 是], [0, 3, 7, 7, 是], [0, 4, 4, 4, 是], [0, 5, 1, 1, 是], [0, 6, 6, 6, 是], [0, 7, 3, 3, 是], [1, 0, 5, 5, 是], [1, 1, 2, 2, 是], [1, 2, 7, 7, 是], [1, 3, 4, 4, 是], [1, 4, 1, 1, 是], [1, 5, 6, 6, 是], [1, 6, 3, 3, 是], [1, 7, 0, 0, 是], [2, 0, 2, 2, 是], [2, 1, 7, 7, 是], [2, 2, 4, 4, 是], [2, 3, 1, 1, 是], [2, 4, 6, 6, 是], [2, 5, 3, 3, 是], [2, 6, 0, 0, 是], [2, 7, 5, 5, 是], [3, 0, 7, 7, 是], [3, 1, 4, 4, 是], [3, 2, 1, 1, 是], [3, 3, 6, 6, 是], [3, 4, 3, 3, 是], [3, 5, 0, 0, 是], [3, 6, 5, 5, 是], [3, 7, 2, 2, 是], [4, 0, 4, 4, 是], [4, 1, 1, 1, 是], [4, 2, 6, 6, 是], [4, 3, 3, 3, 是], [4, 4, 0, 0, 是], [4, 5, 5, 5, 是], [4, 6, 2, 2, 是], [4, 7, 7, 7, 是], [5, 0, 1, 1, 是], [5, 1, 6, 6, 是], [5, 2, 3, 3, 是], [5, 3, 0, 0, 是], [5, 4, 5, 5, 是], [5, 5, 2, 2, 是], [5, 6, 7, 7, 是], [5, 7, 4, 4, 是], [6, 0, 6, 6, 是], [6, 1, 3, 3, 是], [6, 2, 0, 0, 是], [6, 3, 5, 5, 是], [6, 4, 2, 2, 是], [6, 5, 7, 7, 是], [6, 6, 4, 4, 是], [6, 7, 1, 1, 是], [7, 0, 3, 3, 是], [7, 1, 0, 0, 是], [7, 2, 5, 5, 是], [7, 3, 2, 2, 是], [7, 4, 7, 7, 是], [7, 5, 4, 4, 是], [7, 6, 1, 1, 是], [7, 7, 6, 6, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0074/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 75. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-induced-isomorphism-audit；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,7,6,5,4,3,2,1]}。請依 [cosets, image, quotient_table, induced_images, product_law_rows, well_defined, induced_homomorphism, injective, surjective_to_image, certificate] 欄序回答。

查看答案與解釋([[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, 是], [0, 1, 7, 7, 是], [0, 2, 6, 6, 是], [0, 3, 5, 5, 是], [0, 4, 4, 4, 是], [0, 5, 3, 3, 是], [0, 6, 2, 2, 是], [0, 7, 1, 1, 是], [1, 0, 7, 7, 是], [1, 1, 6, 6, 是], [1, 2, 5, 5, 是], [1, 3, 4, 4, 是], [1, 4, 3, 3, 是], [1, 5, 2, 2, 是], [1, 6, 1, 1, 是], [1, 7, 0, 0, 是], [2, 0, 6, 6, 是], [2, 1, 5, 5, 是], [2, 2, 4, 4, 是], [2, 3, 3, 3, 是], [2, 4, 2, 2, 是], [2, 5, 1, 1, 是], [2, 6, 0, 0, 是], [2, 7, 7, 7, 是], [3, 0, 5, 5, 是], [3, 1, 4, 4, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是], [3, 4, 1, 1, 是], [3, 5, 0, 0, 是], [3, 6, 7, 7, 是], [3, 7, 6, 6, 是], [4, 0, 4, 4, 是], [4, 1, 3, 3, 是], [4, 2, 2, 2, 是], [4, 3, 1, 1, 是], [4, 4, 0, 0, 是], [4, 5, 7, 7, 是], [4, 6, 6, 6, 是], [4, 7, 5, 5, 是], [5, 0, 3, 3, 是], [5, 1, 2, 2, 是], [5, 2, 1, 1, 是], [5, 3, 0, 0, 是], [5, 4, 7, 7, 是], [5, 5, 6, 6, 是], [5, 6, 5, 5, 是], [5, 7, 4, 4, 是], [6, 0, 2, 2, 是], [6, 1, 1, 1, 是], [6, 2, 0, 0, 是], [6, 3, 7, 7, 是], [6, 4, 6, 6, 是], [6, 5, 5, 5, 是], [6, 6, 4, 4, 是], [6, 7, 3, 3, 是], [7, 0, 1, 1, 是], [7, 1, 0, 0, 是], [7, 2, 7, 7, 是], [7, 3, 6, 6, 是], [7, 4, 5, 5, 是], [7, 5, 4, 4, 是], [7, 6, 3, 3, 是], [7, 7, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)

Wave6第九層核對=([[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, 是], [0, 1, 7, 7, 是], [0, 2, 6, 6, 是], [0, 3, 5, 5, 是], [0, 4, 4, 4, 是], [0, 5, 3, 3, 是], [0, 6, 2, 2, 是], [0, 7, 1, 1, 是], [1, 0, 7, 7, 是], [1, 1, 6, 6, 是], [1, 2, 5, 5, 是], [1, 3, 4, 4, 是], [1, 4, 3, 3, 是], [1, 5, 2, 2, 是], [1, 6, 1, 1, 是], [1, 7, 0, 0, 是], [2, 0, 6, 6, 是], [2, 1, 5, 5, 是], [2, 2, 4, 4, 是], [2, 3, 3, 3, 是], [2, 4, 2, 2, 是], [2, 5, 1, 1, 是], [2, 6, 0, 0, 是], [2, 7, 7, 7, 是], [3, 0, 5, 5, 是], [3, 1, 4, 4, 是], [3, 2, 3, 3, 是], [3, 3, 2, 2, 是], [3, 4, 1, 1, 是], [3, 5, 0, 0, 是], [3, 6, 7, 7, 是], [3, 7, 6, 6, 是], [4, 0, 4, 4, 是], [4, 1, 3, 3, 是], [4, 2, 2, 2, 是], [4, 3, 1, 1, 是], [4, 4, 0, 0, 是], [4, 5, 7, 7, 是], [4, 6, 6, 6, 是], [4, 7, 5, 5, 是], [5, 0, 3, 3, 是], [5, 1, 2, 2, 是], [5, 2, 1, 1, 是], [5, 3, 0, 0, 是], [5, 4, 7, 7, 是], [5, 5, 6, 6, 是], [5, 6, 5, 5, 是], [5, 7, 4, 4, 是], [6, 0, 2, 2, 是], [6, 1, 1, 1, 是], [6, 2, 0, 0, 是], [6, 3, 7, 7, 是], [6, 4, 6, 6, 是], [6, 5, 5, 5, 是], [6, 6, 4, 4, 是], [6, 7, 3, 3, 是], [7, 0, 1, 1, 是], [7, 1, 0, 0, 是], [7, 2, 7, 7, 是], [7, 3, 6, 6, 是], [7, 4, 5, 5, 是], [7, 5, 4, 4, 是], [7, 6, 3, 3, 是], [7, 7, 2, 2, 是]], 是, 是, 是, 是, induced_map_is_a_bijective_group_homomorphism_onto_image)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0075/)

有限群第一同構定理 · qf-mod46-iso1-induced-isomorphism-audit · candidate





## 76. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1],[1,0]],"source_identity":0,"target_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"target_identity":0,"mapping":[0,2]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2], 4, 2, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2], 4, 2, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0076/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 77. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2],[1,2,0],[2,0,1]],"source_identity":0,"target_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"target_identity":0,"mapping":[0,2,4]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4], 6, 3, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4], 6, 3, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0077/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 78. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"target_identity":0,"mapping":[0,2,4,6]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6], 8, 4, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6], 8, 4, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0078/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 79. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4],[1,2,3,4,0],[2,3,4,0,1],[3,4,0,1,2],[4,0,1,2,3]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"target_identity":0,"mapping":[0,2,4,6,8]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8], 10, 5, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8], 10, 5, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0079/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 80. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"target_identity":0,"mapping":[0,2,4,6,8,10]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10], 12, 6, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10], 12, 6, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0080/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 81. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6],[1,2,3,4,5,6,0],[2,3,4,5,6,0,1],[3,4,5,6,0,1,2],[4,5,6,0,1,2,3],[5,6,0,1,2,3,4],[6,0,1,2,3,4,5]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"target_identity":0,"mapping":[0,2,4,6,8,10,12]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12], 14, 7, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12], 14, 7, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0081/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 82. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3],[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14], 16, 8, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14], 16, 8, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0082/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 83. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8],[1,2,3,4,5,6,7,8,0],[2,3,4,5,6,7,8,0,1],[3,4,5,6,7,8,0,1,2],[4,5,6,7,8,0,1,2,3],[5,6,7,8,0,1,2,3,4],[6,7,8,0,1,2,3,4,5],[7,8,0,1,2,3,4,5,6],[8,0,1,2,3,4,5,6,7]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16], 18, 9, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16], 18, 9, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0083/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 84. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9,0],[2,3,4,5,6,7,8,9,0,1],[3,4,5,6,7,8,9,0,1,2],[4,5,6,7,8,9,0,1,2,3],[5,6,7,8,9,0,1,2,3,4],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6],[8,9,0,1,2,3,4,5,6,7],[9,0,1,2,3,4,5,6,7,8]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 20, 10, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], 20, 10, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0084/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 85. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9,10,0],[2,3,4,5,6,7,8,9,10,0,1],[3,4,5,6,7,8,9,10,0,1,2],[4,5,6,7,8,9,10,0,1,2,3],[5,6,7,8,9,10,0,1,2,3,4],[6,7,8,9,10,0,1,2,3,4,5],[7,8,9,10,0,1,2,3,4,5,6],[8,9,10,0,1,2,3,4,5,6,7],[9,10,0,1,2,3,4,5,6,7,8],[10,0,1,2,3,4,5,6,7,8,9]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 22, 11, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 22, 11, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0085/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 86. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 24, 12, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], 24, 12, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0086/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 87. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9,10,11,12,0],[2,3,4,5,6,7,8,9,10,11,12,0,1],[3,4,5,6,7,8,9,10,11,12,0,1,2],[4,5,6,7,8,9,10,11,12,0,1,2,3],[5,6,7,8,9,10,11,12,0,1,2,3,4],[6,7,8,9,10,11,12,0,1,2,3,4,5],[7,8,9,10,11,12,0,1,2,3,4,5,6],[8,9,10,11,12,0,1,2,3,4,5,6,7],[9,10,11,12,0,1,2,3,4,5,6,7,8],[10,11,12,0,1,2,3,4,5,6,7,8,9],[11,12,0,1,2,3,4,5,6,7,8,9,10],[12,0,1,2,3,4,5,6,7,8,9,10,11]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 26, 13, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], 26, 13, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0087/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 88. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13],[1,2,3,4,5,6,7,8,9,10,11,12,13,0],[2,3,4,5,6,7,8,9,10,11,12,13,0,1],[3,4,5,6,7,8,9,10,11,12,13,0,1,2],[4,5,6,7,8,9,10,11,12,13,0,1,2,3],[5,6,7,8,9,10,11,12,13,0,1,2,3,4],[6,7,8,9,10,11,12,13,0,1,2,3,4,5],[7,8,9,10,11,12,13,0,1,2,3,4,5,6],[8,9,10,11,12,13,0,1,2,3,4,5,6,7],[9,10,11,12,13,0,1,2,3,4,5,6,7,8],[10,11,12,13,0,1,2,3,4,5,6,7,8,9],[11,12,13,0,1,2,3,4,5,6,7,8,9,10],[12,13,0,1,2,3,4,5,6,7,8,9,10,11],[13,0,1,2,3,4,5,6,7,8,9,10,11,12]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 28, 14, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], 28, 14, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0088/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 89. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,0,1,2,3,4,5,6,7,8,9,10,11,12,13]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 30, 15, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], 30, 15, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0089/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 90. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 32, 16, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 32, 16, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0090/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 91. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 34, 17, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], 34, 17, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0091/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 92. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 36, 18, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], 36, 18, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0092/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 93. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 38, 19, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 38, 19, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0093/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 94. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], 40, 20, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0094/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 95. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 42, 21, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], 42, 21, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0095/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 96. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 44, 22, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], 44, 22, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0096/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 97. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],"target_identity":0,"mapping":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([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, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 46, 23, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0097/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 98. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]],"target_identity":0,"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]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 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, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 48, 24, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0098/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 99. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]],"target_identity":0,"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]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 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, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], 50, 25, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0099/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate





## 100. 有限群第一同構定理

完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={"source_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]],"source_identity":0,"target_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44],[46,47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[47,48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46],[48,49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],[49,50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48],[50,51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],[51,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]],"target_identity":0,"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]}。請依 [image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary] 欄序回答。

商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。

查看答案與解釋([0, 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, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)

Wave6第九層核對=([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], 52, 26, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0100/)

有限群第一同構定理 · qf-mod46-iso1-image-codomain-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/validation-report.json)