# 第 24882 題：有限群共軛類 [返回所在分頁 125](https://ocme.evemisslab.com/zh-Hant/questions/page/125/) [← 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-20261114-r03-finite-group-conjugacy-class-foundations/#q-20261015-advance-20261114-r03-finite-group-conjugacy-class-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。 R0048-24882 · q-20261015-advance-20261114-r03-finite-group-conjugacy-class-0082 ## 原題與已知條件 由完整Cayley table導出全部conjugation action與orbits。操作=conjugacy-singleton-nonconjugate-boundary；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":0,"other":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。 - G001 — 由完整Cayley table導出全部conjugation action與orbits。操作=conjugacy-singleton-nonconjugate-boundary；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[1,2,3,4,5,6,7,8,0,10,11,12,13,14,15,16,17,9],[2,3,4,5,6,7,8,0,1,11,12,13,14,15,16,17,9,10],[3,4,5,6,7,8,0,1,2,12,13,14,15,16,17,9,10,11],[4,5,6,7,8,0,1,2,3,13,14,15,16,17,9,10,11,12],[5,6,7,8,0,1,2,3,4,14,15,16,17,9,10,11,12,13],[6,7,8,0,1,2,3,4,5,15,16,17,9,10,11,12,13,14],[7,8,0,1,2,3,4,5,6,16,17,9,10,11,12,13,14,15],[8,0,1,2,3,4,5,6,7,17,9,10,11,12,13,14,15,16],[9,17,16,15,14,13,12,11,10,0,8,7,6,5,4,3,2,1],[10,9,17,16,15,14,13,12,11,1,0,8,7,6,5,4,3,2],[11,10,9,17,16,15,14,13,12,2,1,0,8,7,6,5,4,3],[12,11,10,9,17,16,15,14,13,3,2,1,0,8,7,6,5,4],[13,12,11,10,9,17,16,15,14,4,3,2,1,0,8,7,6,5],[14,13,12,11,10,9,17,16,15,5,4,3,2,1,0,8,7,6],[15,14,13,12,11,10,9,17,16,6,5,4,3,2,1,0,8,7],[16,15,14,13,12,11,10,9,17,7,6,5,4,3,2,1,0,8],[17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0]],"identity":0,"element":0,"other":9}。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-finite-group-conjugacy-class](https://ocme.evemisslab.com/zh-Hant/mko/mko-finite-group-conjugacy-class/) ## 逐步推導 S001 · implies操作前 source roles = [conjugacy-singleton-nonconjugate-boundary] 操作後 derived = [[element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary]] 條件：無 直接列來源完整欄序，不需JavaScript；原題、布林旗標、數學物件與非聲稱保持分層。 步驟識別與依賴規則：source_ordered_fields_without_JS · 依賴：G001 S002 · implies操作前 source roles = [Conjugacy ORIGINAL group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0] 操作後 derived = [18, [[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], [16, 16, 16, 16], [17, 17, 17, 17]], 是] 條件：無 先驗完整n×n閉合表與實際identity標籤，非預設0。mod44上限32，mod45上限64；同一carrier不能混入另一表。 步驟識別與依賴規則：Group_complete_Cayley_signature_actual_identity · 依賴：G001, S001 S003 · implies操作前 source roles = [Conjugacy ORIGINAL group, 18, FOR EVERY x,y,z in0..n-1] 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, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 5832, 是] 條件：無 矩陣(x,y)逐全部z核(xy)z=x(yz)，故涵蓋全部n³三元組，不是抽樣。表已在前步完整列出，不重印數百萬同義查表項。 步驟識別與依賴規則：Group_ALL_associativity_triples_compact_complete_pair_matrix · 依賴：G001, S002 S004 · implies操作前 source roles = [Conjugacy ORIGINAL group, 0] 操作後 derived = [[0, 8, 7, 6, 5, 4, 3, 2, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17], [[0, 0, 0, 0], [1, 8, 0, 0], [2, 7, 0, 0], [3, 6, 0, 0], [4, 5, 0, 0], [5, 4, 0, 0], [6, 3, 0, 0], [7, 2, 0, 0], [8, 1, 0, 0], [9, 9, 0, 0], [10, 10, 0, 0], [11, 11, 0, 0], [12, 12, 0, 0], [13, 13, 0, 0], [14, 14, 0, 0], [15, 15, 0, 0], [16, 16, 0, 0], [17, 17, 0, 0]], 是] 條件：無 每個元素的左、右逆都要回到同一實際identity；結合律使兩側逆唯一。這才完成群前提。 步驟識別與依賴規則：Group_ALL_two_sided_inverses · 依賴：G001, S003 S005 · implies操作前 source roles = [0, 0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S004 S006 · implies操作前 source roles = [1, 8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 9, 10]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S005 S007 · implies操作前 source roles = [2, 7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 9, 10, 11, 12]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S006 S008 · implies操作前 source roles = [3, 6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 15, 16, 17, 9, 10, 11, 12, 13, 14]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S007 S009 · implies操作前 source roles = [4, 5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 9, 10, 11, 12, 13, 14, 15, 16]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S008 S010 · implies操作前 source roles = [5, 4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 9]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S009 S011 · implies操作前 source roles = [6, 3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 9, 10, 11]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S010 S012 · implies操作前 source roles = [7, 2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17, 9, 10, 11, 12, 13]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S011 S013 · implies操作前 source roles = [8, 1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 16, 17, 9, 10, 11, 12, 13, 14, 15]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S012 S014 · implies操作前 source roles = [9, 9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [0, 8, 7, 6, 5, 4, 3, 2, 1, 9, 17, 16, 15, 14, 13, 12, 11, 10]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S013 S015 · implies操作前 source roles = [10, 10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [0, 8, 7, 6, 5, 4, 3, 2, 1, 11, 10, 9, 17, 16, 15, 14, 13, 12]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S014 S016 · implies操作前 source roles = [11, 11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [0, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 17, 16, 15, 14]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S015 S017 · implies操作前 source roles = [12, 12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [0, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 10, 9, 17, 16]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S016 S018 · implies操作前 source roles = [13, 13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [0, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 10, 9]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S017 S019 · implies操作前 source roles = [14, 14, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [0, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 17, 16, 15, 14, 13, 12, 11]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S018 S020 · implies操作前 source roles = [15, 15, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [0, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 17, 16, 15, 14, 13]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S019 S021 · implies操作前 source roles = [16, 16, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [0, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 17, 16, 15]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S020 S022 · implies操作前 source roles = [17, 17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]] 操作後 derived = [[17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [0, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 17]] 條件：無 對每個原元素x先算gx再乘g^-1，形成完整acting row；這是element action，不是conjugate subgroup。 步驟識別與依賴規則：Conjugacy_EVERY_g_EVERY_element_exact_action_row · 依賴：G001, S021 S023 · implies操作前 source roles = [0, 18, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 15, 16, 17, 9, 10, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 16, 17, 9, 10, 11, 12, 13, 14, 15], [0, 8, 7, 6, 5, 4, 3, 2, 1, 9, 17, 16, 15, 14, 13, 12, 11, 10], [0, 8, 7, 6, 5, 4, 3, 2, 1, 11, 10, 9, 17, 16, 15, 14, 13, 12], [0, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 17, 16, 15, 14], [0, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 10, 9, 17, 16], [0, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 10, 9], [0, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 17, 16, 15, 14, 13, 12, 11], [0, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 17, 16, 15, 14, 13], [0, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 17, 16, 15], [0, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 17]]] 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, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 是, (gh)x(gh)^-1=g(hxh^-1)g^-1] 條件：無 逐全部g,h,x核composition，亦由(gh)^-1=h^-1g^-1及結合律推同一公式；identity另驗。不是僅列出一個action表就假定axioms。 步驟識別與依賴規則：Conjugacy_ALL_action_axioms_and_GENERAL_algebra · 依賴：G001, S022 S024 · implies操作前 source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 15, 16, 17, 9, 10, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 16, 17, 9, 10, 11, 12, 13, 14, 15], [0, 8, 7, 6, 5, 4, 3, 2, 1, 9, 17, 16, 15, 14, 13, 12, 11, 10], [0, 8, 7, 6, 5, 4, 3, 2, 1, 11, 10, 9, 17, 16, 15, 14, 13, 12], [0, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 17, 16, 15, 14], [0, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 10, 9, 17, 16], [0, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 10, 9], [0, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 17, 16, 15, 14, 13, 12, 11], [0, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 17, 16, 15, 14, 13], [0, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 17, 16, 15], [0, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 17]]] 操作後 derived = [[[0], [1, 8], [2, 7], [3, 6], [4, 5], [9, 10, 11, 12, 13, 14, 15, 16, 17]], 18, 18, 是] 條件：無 e給自反、g^-1給反向、群乘積給傳遞；故所有完整orbits相等或不交並覆蓋G。按來源coset-like列表排序不代表它們是cosets；不引centralizer/class-size或類方程。 步驟識別與依賴規則：Conjugacy_complete_orbits_equal_or_disjoint_partition · 依賴：G001, S023 S025 · implies操作前 source roles = [0] 操作後 derived = [[[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0]], [0]] 條件：無 保每個acting g的像，再去重成排序的完整class。 步驟識別與依賴規則：Conjugacy_actual_element_ALL_images_not_coset · 依賴：G001, S024 S026 · implies操作前 source roles = [0, 9, [0]] 操作後 derived = [[[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], [8, 8, 8, 是], [9, 9, 9, 是], [10, 10, 10, 是], [11, 11, 11, 是], [12, 12, 12, 是], [13, 13, 13, 是], [14, 14, 14, 是], [15, 15, 15, 是], [16, 16, 16, 是], [17, 17, 17, 是]], 是, 否, 是] 條件：無 gxg^-1=x右乘g等價gx=xg，故singleton恰當全g commute。other真不在完整orbit才可給nonconjugate證書，不聲稱所有class單點。 步驟識別與依賴規則：Conjugacy_singleton_iff_ALL_commute_and_actual_other_outside · 依賴：G001, S025 ## 逐步反向核對 - S026 · 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。 - S025 · 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。 - S024 · 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。 - S023 · 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。 - S022 · 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。 - S021 · 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。 - S020 · 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。 - S019 · 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。 - S018 · 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。 - S017 · 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。 - S016 · 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。 - S015 · 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。 - S014 · 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。 - S013 · 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。 - S012 · 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。 - 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, [0], 是, 9, 否, 是, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula) 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: 2006831324b048f88437bcd0b8e2e4c4d8e9d5b479385afa0a696ce4fa6bc5c0 - candidate record SHA-256: 535ec97ab1bbaff7346ba7cdec54d39696ba1de983ccb174e382fce1589bdb55 - semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0 - question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9 - Lean: NOT_RUN · human review: NOT_PERFORMED [同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r03-finite-group-conjugacy-class-0082.json)