第 24524 題：有限正規子群

原題與已知條件

完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；資料={"group_table":[[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],[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]],"identity":4,"subset":[0,1,2,3,4,5,6,7,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-conjugate-set-audit；資料={"group_table":[[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],[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]],"identity":4,"subset":[0,1,2,3,4,5,6,7,8,9]}。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-normal-subgroup

逐步推導

source roles = [normal-conjugate-set-audit]

derived = [[subgroup, conjugate_rows, normal, certificate]]

source roles = [Normality ORIGINAL group, 32, [[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], [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]], 4]

derived = [10, [[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]], 是]

source roles = [Normality ORIGINAL group, 10, FOR EVERY x,y,z in0..n-1]

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]], 1000, 是]

source roles = [Normality ORIGINAL group, 4]

derived = [[8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [[0, 8, 4, 4], [1, 7, 4, 4], [2, 6, 4, 4], [3, 5, 4, 4], [4, 4, 4, 4], [5, 3, 4, 4], [6, 2, 4, 4], [7, 1, 4, 4], [8, 0, 4, 4], [9, 9, 4, 4]], 是]

source roles = [Normality ORIGINAL H, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 4]

derived = [是, [[0, 0, 6, 是], [0, 1, 7, 是], [0, 2, 8, 是], [0, 3, 9, 是], [0, 4, 0, 是], [0, 5, 1, 是], [0, 6, 2, 是], [0, 7, 3, 是], [0, 8, 4, 是], [0, 9, 5, 是], [1, 0, 7, 是], [1, 1, 8, 是], [1, 2, 9, 是], [1, 3, 0, 是], [1, 4, 1, 是], [1, 5, 2, 是], [1, 6, 3, 是], [1, 7, 4, 是], [1, 8, 5, 是], [1, 9, 6, 是], [2, 0, 8, 是], [2, 1, 9, 是], [2, 2, 0, 是], [2, 3, 1, 是], [2, 4, 2, 是], [2, 5, 3, 是], [2, 6, 4, 是], [2, 7, 5, 是], [2, 8, 6, 是], [2, 9, 7, 是], [3, 0, 9, 是], [3, 1, 0, 是], [3, 2, 1, 是], [3, 3, 2, 是], [3, 4, 3, 是], [3, 5, 4, 是], [3, 6, 5, 是], [3, 7, 6, 是], [3, 8, 7, 是], [3, 9, 8, 是], [4, 0, 0, 是], [4, 1, 1, 是], [4, 2, 2, 是], [4, 3, 3, 是], [4, 4, 4, 是], [4, 5, 5, 是], [4, 6, 6, 是], [4, 7, 7, 是], [4, 8, 8, 是], [4, 9, 9, 是], [5, 0, 1, 是], [5, 1, 2, 是], [5, 2, 3, 是], [5, 3, 4, 是], [5, 4, 5, 是], [5, 5, 6, 是], [5, 6, 7, 是], [5, 7, 8, 是], [5, 8, 9, 是], [5, 9, 0, 是], [6, 0, 2, 是], [6, 1, 3, 是], [6, 2, 4, 是], [6, 3, 5, 是], [6, 4, 6, 是], [6, 5, 7, 是], [6, 6, 8, 是], [6, 7, 9, 是], [6, 8, 0, 是], [6, 9, 1, 是], [7, 0, 3, 是], [7, 1, 4, 是], [7, 2, 5, 是], [7, 3, 6, 是], [7, 4, 7, 是], [7, 5, 8, 是], [7, 6, 9, 是], [7, 7, 0, 是], [7, 8, 1, 是], [7, 9, 2, 是], [8, 0, 4, 是], [8, 1, 5, 是], [8, 2, 6, 是], [8, 3, 7, 是], [8, 4, 8, 是], [8, 5, 9, 是], [8, 6, 0, 是], [8, 7, 1, 是], [8, 8, 2, 是], [8, 9, 3, 是], [9, 0, 5, 是], [9, 1, 6, 是], [9, 2, 7, 是], [9, 3, 8, 是], [9, 4, 9, 是], [9, 5, 0, 是], [9, 6, 1, 是], [9, 7, 2, 是], [9, 8, 3, 是], [9, 9, 4, 是]], 是, [[0, 8, 是], [1, 7, 是], [2, 6, 是], [3, 5, 是], [4, 4, 是], [5, 3, 是], [6, 2, 是], [7, 1, 是], [8, 0, 是], [9, 9, 是]], 是, 是]

source roles = [Normality ORIGINAL H, 0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 8]

derived = [[[0, 6, 8, 0], [1, 7, 8, 1], [2, 8, 8, 2], [3, 9, 8, 3], [4, 0, 8, 4], [5, 1, 8, 5], [6, 2, 8, 6], [7, 3, 8, 7], [8, 4, 8, 8], [9, 5, 8, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 7]

derived = [[[0, 7, 7, 0], [1, 8, 7, 1], [2, 9, 7, 2], [3, 0, 7, 3], [4, 1, 7, 4], [5, 2, 7, 5], [6, 3, 7, 6], [7, 4, 7, 7], [8, 5, 7, 8], [9, 6, 7, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 6]

derived = [[[0, 8, 6, 0], [1, 9, 6, 1], [2, 0, 6, 2], [3, 1, 6, 3], [4, 2, 6, 4], [5, 3, 6, 5], [6, 4, 6, 6], [7, 5, 6, 7], [8, 6, 6, 8], [9, 7, 6, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 5]

derived = [[[0, 9, 5, 0], [1, 0, 5, 1], [2, 1, 5, 2], [3, 2, 5, 3], [4, 3, 5, 4], [5, 4, 5, 5], [6, 5, 5, 6], [7, 6, 5, 7], [8, 7, 5, 8], [9, 8, 5, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 4]

derived = [[[0, 0, 4, 0], [1, 1, 4, 1], [2, 2, 4, 2], [3, 3, 4, 3], [4, 4, 4, 4], [5, 5, 4, 5], [6, 6, 4, 6], [7, 7, 4, 7], [8, 8, 4, 8], [9, 9, 4, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 3]

derived = [[[0, 1, 3, 0], [1, 2, 3, 1], [2, 3, 3, 2], [3, 4, 3, 3], [4, 5, 3, 4], [5, 6, 3, 5], [6, 7, 3, 6], [7, 8, 3, 7], [8, 9, 3, 8], [9, 0, 3, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 2]

derived = [[[0, 2, 2, 0], [1, 3, 2, 1], [2, 4, 2, 2], [3, 5, 2, 3], [4, 6, 2, 4], [5, 7, 2, 5], [6, 8, 2, 6], [7, 9, 2, 7], [8, 0, 2, 8], [9, 1, 2, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 1]

derived = [[[0, 3, 1, 0], [1, 4, 1, 1], [2, 5, 1, 2], [3, 6, 1, 3], [4, 7, 1, 4], [5, 8, 1, 5], [6, 9, 1, 6], [7, 0, 1, 7], [8, 1, 1, 8], [9, 2, 1, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 0]

derived = [[[0, 4, 0, 0], [1, 5, 0, 1], [2, 6, 0, 2], [3, 7, 0, 3], [4, 8, 0, 4], [5, 9, 0, 5], [6, 0, 0, 6], [7, 1, 0, 7], [8, 2, 0, 8], [9, 3, 0, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

source roles = [Normality ORIGINAL H, 9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 9]

derived = [[[0, 5, 9, 0], [1, 6, 9, 1], [2, 7, 9, 2], [3, 8, 9, 3], [4, 9, 9, 4], [5, 0, 9, 5], [6, 1, 9, 6], [7, 2, 9, 7], [8, 3, 9, 8], [9, 4, 9, 9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]

逐步反向核對

• 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, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, every_group_element_conjugates_H_to_itself)

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。