# 第 25732 題：有限群第一同構定理

[返回所在分頁 129](https://ocme.evemisslab.com/zh-Hant/questions/page/129/)

[← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/#q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0082)

本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。

R0050-25732 · q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0082



## 原題與已知條件

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 exact_ordered_tuple_declared_by_interface 欄序回答。


- G001 — 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 exact_ordered_tuple_declared_by_interface 欄序回答。





## 定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。


- [mko-first-isomorphism-theorem-finite-groups](https://ocme.evemisslab.com/zh-Hant/mko/mko-first-isomorphism-theorem-finite-groups/)





## 逐步推導

S001 · implies操作前

source roles = [iso1-image-codomain-boundary]

操作後

derived = [[image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary]]

條件：無

直接列完整來源欄序，不需JavaScript。LP weak為13欄、zero與strong為15欄，前提與角色依原operation／mode，不補欄。

步驟識別與依賴規則：source_ordered_fields_without_JS · 依賴：G001





S002 · implies操作前

source roles = [Iso1 ORIGINAL SOURCE, [[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]

操作後

derived = [8, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7]], 是]

條件：無

完整closed Cayley table與實際identity先資格核對，群階可到64，不假設identity0。

步驟識別與依賴規則：Group_complete_table_actual_identity_bound64 · 依賴：G001, S001





S003 · implies操作前

source roles = [Iso1 ORIGINAL SOURCE, 8, 0]

操作後

derived = [[[是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是]], 512, [0, 7, 6, 5, 4, 3, 2, 1]]

條件：無

每pair格逐全部第三元素核結合律，完整n³而非抽樣；inverse同時左右回實際identity。

步驟識別與依賴規則：Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · 依賴：G001, S002





S004 · implies操作前

source roles = [Iso1 ORIGINAL TARGET, [[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]], 0]

操作後

derived = [16, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15]], 是]

條件：無

完整closed Cayley table與實際identity先資格核對，群階可到64，不假設identity0。

步驟識別與依賴規則：Group_complete_table_actual_identity_bound64 · 依賴：G001, S003





S005 · implies操作前

source roles = [Iso1 ORIGINAL TARGET, 16, 0]

操作後

derived = [[[是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 4096, [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]

條件：無

每pair格逐全部第三元素核結合律，完整n³而非抽樣；inverse同時左右回實際identity。

步驟識別與依賴規則：Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · 依賴：G001, S004





S006 · implies操作前

source roles = [Iso1 ACTUAL kernel, [0], 0]

操作後

derived = [是, [[0, 0, 0, 是]], 是, [[0, 0, 是]], 是, 是, [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是]], 是]

條件：無

含identity、全部product與inverse closure各核，三者合取才成subgroup；再逐全部g驗normal。即使某map失敗，這些實際property仍可分別True，不由定理名稱填旗標。

步驟識別與依賴規則：Subgroup_actual_identity_product_inverse_ALL_conjugates · 依賴：G001, S005





S007 · implies操作前

source roles = [Iso1 ACTUAL image, [0, 2, 4, 6, 8, 10, 12, 14], 0]

操作後

derived = [是, [[0, 0, 0, 是], [0, 2, 2, 是], [0, 4, 4, 是], [0, 6, 6, 是], [0, 8, 8, 是], [0, 10, 10, 是], [0, 12, 12, 是], [0, 14, 14, 是], [2, 0, 2, 是], [2, 2, 4, 是], [2, 4, 6, 是], [2, 6, 8, 是], [2, 8, 10, 是], [2, 10, 12, 是], [2, 12, 14, 是], [2, 14, 0, 是], [4, 0, 4, 是], [4, 2, 6, 是], [4, 4, 8, 是], [4, 6, 10, 是], [4, 8, 12, 是], [4, 10, 14, 是], [4, 12, 0, 是], [4, 14, 2, 是], [6, 0, 6, 是], [6, 2, 8, 是], [6, 4, 10, 是], [6, 6, 12, 是], [6, 8, 14, 是], [6, 10, 0, 是], [6, 12, 2, 是], [6, 14, 4, 是], [8, 0, 8, 是], [8, 2, 10, 是], [8, 4, 12, 是], [8, 6, 14, 是], [8, 8, 0, 是], [8, 10, 2, 是], [8, 12, 4, 是], [8, 14, 6, 是], [10, 0, 10, 是], [10, 2, 12, 是], [10, 4, 14, 是], [10, 6, 0, 是], [10, 8, 2, 是], [10, 10, 4, 是], [10, 12, 6, 是], [10, 14, 8, 是], [12, 0, 12, 是], [12, 2, 14, 是], [12, 4, 0, 是], [12, 6, 2, 是], [12, 8, 4, 是], [12, 10, 6, 是], [12, 12, 8, 是], [12, 14, 10, 是], [14, 0, 14, 是], [14, 2, 0, 是], [14, 4, 2, 是], [14, 6, 4, 是], [14, 8, 6, 是], [14, 10, 8, 是], [14, 12, 10, 是], [14, 14, 12, 是]], 是, [[0, 0, 是], [2, 14, 是], [4, 12, 是], [6, 10, 是], [8, 8, 是], [10, 6, 是], [12, 4, 是], [14, 2, 是]], 是, 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14], 是], [1, [0, 2, 4, 6, 8, 10, 12, 14], 是], [2, [0, 2, 4, 6, 8, 10, 12, 14], 是], [3, [0, 2, 4, 6, 8, 10, 12, 14], 是], [4, [0, 2, 4, 6, 8, 10, 12, 14], 是], [5, [0, 2, 4, 6, 8, 10, 12, 14], 是], [6, [0, 2, 4, 6, 8, 10, 12, 14], 是], [7, [0, 2, 4, 6, 8, 10, 12, 14], 是], [8, [0, 2, 4, 6, 8, 10, 12, 14], 是], [9, [0, 2, 4, 6, 8, 10, 12, 14], 是], [10, [0, 2, 4, 6, 8, 10, 12, 14], 是], [11, [0, 2, 4, 6, 8, 10, 12, 14], 是], [12, [0, 2, 4, 6, 8, 10, 12, 14], 是], [13, [0, 2, 4, 6, 8, 10, 12, 14], 是], [14, [0, 2, 4, 6, 8, 10, 12, 14], 是], [15, [0, 2, 4, 6, 8, 10, 12, 14], 是]], 是]

條件：無

含identity、全部product與inverse closure各核，三者合取才成subgroup；再逐全部g驗normal。即使某map失敗，這些實際property仍可分別True，不由定理名稱填旗標。

步驟識別與依賴規則：Subgroup_actual_identity_product_inverse_ALL_conjugates · 依賴：G001, S006





S008 · implies操作前

source roles = [[0, 2, 4, 6, 8, 10, 12, 14], 0, 0]

操作後

derived = [[[0, 0, 0, 0, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 6, 6, 是], [0, 4, 8, 8, 是], [0, 5, 10, 10, 是], [0, 6, 12, 12, 是], [0, 7, 14, 14, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 6, 6, 是], [1, 3, 8, 8, 是], [1, 4, 10, 10, 是], [1, 5, 12, 12, 是], [1, 6, 14, 14, 是], [1, 7, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 6, 6, 是], [2, 2, 8, 8, 是], [2, 3, 10, 10, 是], [2, 4, 12, 12, 是], [2, 5, 14, 14, 是], [2, 6, 0, 0, 是], [2, 7, 2, 2, 是], [3, 0, 6, 6, 是], [3, 1, 8, 8, 是], [3, 2, 10, 10, 是], [3, 3, 12, 12, 是], [3, 4, 14, 14, 是], [3, 5, 0, 0, 是], [3, 6, 2, 2, 是], [3, 7, 4, 4, 是], [4, 0, 8, 8, 是], [4, 1, 10, 10, 是], [4, 2, 12, 12, 是], [4, 3, 14, 14, 是], [4, 4, 0, 0, 是], [4, 5, 2, 2, 是], [4, 6, 4, 4, 是], [4, 7, 6, 6, 是], [5, 0, 10, 10, 是], [5, 1, 12, 12, 是], [5, 2, 14, 14, 是], [5, 3, 0, 0, 是], [5, 4, 2, 2, 是], [5, 5, 4, 4, 是], [5, 6, 6, 6, 是], [5, 7, 8, 8, 是], [6, 0, 12, 12, 是], [6, 1, 14, 14, 是], [6, 2, 0, 0, 是], [6, 3, 2, 2, 是], [6, 4, 4, 4, 是], [6, 5, 6, 6, 是], [6, 6, 8, 8, 是], [6, 7, 10, 10, 是], [7, 0, 14, 14, 是], [7, 1, 0, 0, 是], [7, 2, 2, 2, 是], [7, 3, 4, 4, 是], [7, 4, 6, 6, 是], [7, 5, 8, 8, 是], [7, 6, 10, 10, 是], [7, 7, 12, 12, 是]], 是, null, [0], [0, 2, 4, 6, 8, 10, 12, 14], 是, 是]

條件：無

全pair驗同態，並取完整target identity preimage與image。qualification允許bad map且property按實際核，可能同態False而kernel normal／image subgroup True；空kernel不補identity fallback。

步驟識別與依賴規則：Iso1_ALL_product_laws_ACTUAL_kernel_image_properties_before_theorem · 依賴：G001, S007





S009 · implies操作前

source roles = [[0], 是, [0, 2, 4, 6, 8, 10, 12, 14]]

操作後

derived = [[[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, 0, [0], 是], [0, 1, [1], 是], [0, 2, [2], 是], [0, 3, [3], 是], [0, 4, [4], 是], [0, 5, [5], 是], [0, 6, [6], 是], [0, 7, [7], 是], [1, 0, [1], 是], [1, 1, [2], 是], [1, 2, [3], 是], [1, 3, [4], 是], [1, 4, [5], 是], [1, 5, [6], 是], [1, 6, [7], 是], [1, 7, [0], 是], [2, 0, [2], 是], [2, 1, [3], 是], [2, 2, [4], 是], [2, 3, [5], 是], [2, 4, [6], 是], [2, 5, [7], 是], [2, 6, [0], 是], [2, 7, [1], 是], [3, 0, [3], 是], [3, 1, [4], 是], [3, 2, [5], 是], [3, 3, [6], 是], [3, 4, [7], 是], [3, 5, [0], 是], [3, 6, [1], 是], [3, 7, [2], 是], [4, 0, [4], 是], [4, 1, [5], 是], [4, 2, [6], 是], [4, 3, [7], 是], [4, 4, [0], 是], [4, 5, [1], 是], [4, 6, [2], 是], [4, 7, [3], 是], [5, 0, [5], 是], [5, 1, [6], 是], [5, 2, [7], 是], [5, 3, [0], 是], [5, 4, [1], 是], [5, 5, [2], 是], [5, 6, [3], 是], [5, 7, [4], 是], [6, 0, [6], 是], [6, 1, [7], 是], [6, 2, [0], 是], [6, 3, [1], 是], [6, 4, [2], 是], [6, 5, [3], 是], [6, 6, [4], 是], [6, 7, [5], 是], [7, 0, [7], 是], [7, 1, [0], 是], [7, 2, [1], 是], [7, 3, [2], 是], [7, 4, [3], 是], [7, 5, [4], 是], [7, 6, [5], 是], [7, 7, [6], 是]], [[0, [0], [0], 是], [1, [1], [2], 是], [2, [2], [4], 是], [3, [3], [6], 是], [4, [4], [8], 是], [5, [5], [10], 是], [6, [6], [12], 是], [7, [7], [14], 是]], 是]

條件：無

kernel normal使(ah)(bk)=ab(b^-1hb)k∈abK，逐全部代表元驗product coset相同；同一coset的每個f(g)也須同值才定義bar_f。carrier是coset SETS而非首代表元。

步驟識別與依賴規則：Iso1_full_LEFT_coset_quotient_ALL_products_and_representative_images · 依賴：G001, S008





S010 · implies操作前

source roles = [[[0], [1], [2], [3], [4], [5], [6], [7]], [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14]]

操作後

derived = [[[0, 0, 0, 0, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 6, 6, 是], [0, 4, 8, 8, 是], [0, 5, 10, 10, 是], [0, 6, 12, 12, 是], [0, 7, 14, 14, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 6, 6, 是], [1, 3, 8, 8, 是], [1, 4, 10, 10, 是], [1, 5, 12, 12, 是], [1, 6, 14, 14, 是], [1, 7, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 6, 6, 是], [2, 2, 8, 8, 是], [2, 3, 10, 10, 是], [2, 4, 12, 12, 是], [2, 5, 14, 14, 是], [2, 6, 0, 0, 是], [2, 7, 2, 2, 是], [3, 0, 6, 6, 是], [3, 1, 8, 8, 是], [3, 2, 10, 10, 是], [3, 3, 12, 12, 是], [3, 4, 14, 14, 是], [3, 5, 0, 0, 是], [3, 6, 2, 2, 是], [3, 7, 4, 4, 是], [4, 0, 8, 8, 是], [4, 1, 10, 10, 是], [4, 2, 12, 12, 是], [4, 3, 14, 14, 是], [4, 4, 0, 0, 是], [4, 5, 2, 2, 是], [4, 6, 4, 4, 是], [4, 7, 6, 6, 是], [5, 0, 10, 10, 是], [5, 1, 12, 12, 是], [5, 2, 14, 14, 是], [5, 3, 0, 0, 是], [5, 4, 2, 2, 是], [5, 5, 4, 4, 是], [5, 6, 6, 6, 是], [5, 7, 8, 8, 是], [6, 0, 12, 12, 是], [6, 1, 14, 14, 是], [6, 2, 0, 0, 是], [6, 3, 2, 2, 是], [6, 4, 4, 4, 是], [6, 5, 6, 6, 是], [6, 6, 8, 8, 是], [6, 7, 10, 10, 是], [7, 0, 14, 14, 是], [7, 1, 0, 0, 是], [7, 2, 2, 2, 是], [7, 3, 4, 4, 是], [7, 4, 6, 6, 是], [7, 5, 8, 8, 是], [7, 6, 10, 10, 是], [7, 7, 12, 12, 是]], 是, 是, 是, 是, 否]

條件：無

bar_f保乘法由f(ab)=f(a)f(b)；若f(a)=f(b)，a^-1b∈K故同coset，給injective。任意image值f(g)由gK命中，給onto IMAGE；不由同cardinality代替這些責任，也不自動onto整個target。

步驟識別與依賴規則：Iso1_ALL_induced_product_pairs_injective_and_onto_IMAGE_separate · 依賴：G001, S009





S011 · implies操作前

source roles = [[0, 2, 4, 6, 8, 10, 12, 14], 16, 8]

操作後

derived = [否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否]

條件：無

只有實際f滿射，image才等同整個codomain。最後No保一般抽象證明／形式化責任，不否定本題完整有限同構證書或教學理由。

步驟識別與依賴規則：Iso1_image_VS_codomain_scope_and_finite_general_proof_boundary · 依賴：G001, S010









## 逐步反向核對


- S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。





## 結論

([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)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。





## 本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。

來源雜湊與覆核憑證
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 74d1960bd7d34043225b3ca55d365c454448e2c0b370989490db8804f2bb31f9
- candidate record SHA-256: f9ba87b60b1bd072b285c84c62480cd118b3caf6142a07a7a3ad6ce528a1247e
- semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82
- question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f
- Lean: NOT_RUN · human review: NOT_PERFORMED

[同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0082.json)