第 25355 題：有限商群

原題與已知條件

完整有限群、subgroup與全部coset partition及代表元責任。操作=quotient-normal-coset-partition；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"identity":0,"subset":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整有限群、subgroup與全部coset partition及代表元責任。操作=quotient-normal-coset-partition；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],"identity":0,"subset":[0,2,4,6,8,10]}。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-quotient-group

逐步推導

source roles = [quotient-normal-coset-partition]

derived = [[subgroup, normal, cosets, partition_certificate, coset_count, certificate]]

source roles = [[[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]], 0]

derived = [12, [[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]], 是]

source roles = [12, 0]

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]], 1728, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]

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

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

source roles = [[0, 2, 4, 6, 8, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]]

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

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

derived = [[[0, 0, 0, 0], [0, 2, 2, 0], [0, 4, 4, 0], [0, 6, 6, 0], [0, 8, 8, 0], [0, 10, 10, 0], [2, 0, 2, 0], [2, 2, 4, 0], [2, 4, 6, 0], [2, 6, 8, 0], [2, 8, 10, 0], [2, 10, 0, 0], [4, 0, 4, 0], [4, 2, 6, 0], [4, 4, 8, 0], [4, 6, 10, 0], [4, 8, 0, 0], [4, 10, 2, 0], [6, 0, 6, 0], [6, 2, 8, 0], [6, 4, 10, 0], [6, 6, 0, 0], [6, 8, 2, 0], [6, 10, 4, 0], [8, 0, 8, 0], [8, 2, 10, 0], [8, 4, 0, 0], [8, 6, 2, 0], [8, 8, 4, 0], [8, 10, 6, 0], [10, 0, 10, 0], [10, 2, 0, 0], [10, 4, 2, 0], [10, 6, 4, 0], [10, 8, 6, 0], [10, 10, 8, 0]], [0], 是]

source roles = [0, 1, [0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]]

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

source roles = [1, 0, [1, 3, 5, 7, 9, 11], [0, 2, 4, 6, 8, 10]]

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

source roles = [1, 1, [1, 3, 5, 7, 9, 11], [1, 3, 5, 7, 9, 11]]

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

source roles = [[0, 2, 4, 6, 8, 10], 是, [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]]]

derived = [[[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], 是, 否]

逐步反向核對

• 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], 是, [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 是, 2, quotient_elements_are_disjoint_complete_coset_sets_not_representatives)

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。