第 25310 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-coset-multiplication-table；資料={"group_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]],"identity":0,"subset":[0,2,4,6,8,10,12,14,16,18,20]}。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-coset-multiplication-table；資料={"group_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]],"identity":0,"subset":[0,2,4,6,8,10,12,14,16,18,20]}。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-coset-multiplication-table]

derived = [[cosets, multiplication_table, identity_coset, inverse_rows, well_defined, commutativity_certificate, certificate]]

source roles = [[[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]], 0]

derived = [22, [[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], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21]], 是]

source roles = [22, 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, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 10648, [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]

source roles = [0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]]

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], [0, 12, 12, 是], [0, 14, 14, 是], [0, 16, 16, 是], [0, 18, 18, 是], [0, 20, 20, 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, 16, 是], [2, 16, 18, 是], [2, 18, 20, 是], [2, 20, 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, 16, 是], [4, 14, 18, 是], [4, 16, 20, 是], [4, 18, 0, 是], [4, 20, 2, 是], [6, 0, 6, 是], [6, 2, 8, 是], [6, 4, 10, 是], [6, 6, 12, 是], [6, 8, 14, 是], [6, 10, 16, 是], [6, 12, 18, 是], [6, 14, 20, 是], [6, 16, 0, 是], [6, 18, 2, 是], [6, 20, 4, 是], [8, 0, 8, 是], [8, 2, 10, 是], [8, 4, 12, 是], [8, 6, 14, 是], [8, 8, 16, 是], [8, 10, 18, 是], [8, 12, 20, 是], [8, 14, 0, 是], [8, 16, 2, 是], [8, 18, 4, 是], [8, 20, 6, 是], [10, 0, 10, 是], [10, 2, 12, 是], [10, 4, 14, 是], [10, 6, 16, 是], [10, 8, 18, 是], [10, 10, 20, 是], [10, 12, 0, 是], [10, 14, 2, 是], [10, 16, 4, 是], [10, 18, 6, 是], [10, 20, 8, 是], [12, 0, 12, 是], [12, 2, 14, 是], [12, 4, 16, 是], [12, 6, 18, 是], [12, 8, 20, 是], [12, 10, 0, 是], [12, 12, 2, 是], [12, 14, 4, 是], [12, 16, 6, 是], [12, 18, 8, 是], [12, 20, 10, 是], [14, 0, 14, 是], [14, 2, 16, 是], [14, 4, 18, 是], [14, 6, 20, 是], [14, 8, 0, 是], [14, 10, 2, 是], [14, 12, 4, 是], [14, 14, 6, 是], [14, 16, 8, 是], [14, 18, 10, 是], [14, 20, 12, 是], [16, 0, 16, 是], [16, 2, 18, 是], [16, 4, 20, 是], [16, 6, 0, 是], [16, 8, 2, 是], [16, 10, 4, 是], [16, 12, 6, 是], [16, 14, 8, 是], [16, 16, 10, 是], [16, 18, 12, 是], [16, 20, 14, 是], [18, 0, 18, 是], [18, 2, 20, 是], [18, 4, 0, 是], [18, 6, 2, 是], [18, 8, 4, 是], [18, 10, 6, 是], [18, 12, 8, 是], [18, 14, 10, 是], [18, 16, 12, 是], [18, 18, 14, 是], [18, 20, 16, 是], [20, 0, 20, 是], [20, 2, 0, 是], [20, 4, 2, 是], [20, 6, 4, 是], [20, 8, 6, 是], [20, 10, 8, 是], [20, 12, 10, 是], [20, 14, 12, 是], [20, 16, 14, 是], [20, 18, 16, 是], [20, 20, 18, 是]], [[0, 0, 是], [2, 20, 是], [4, 18, 是], [6, 16, 是], [8, 14, 是], [10, 12, 是], [12, 10, 是], [14, 8, 是], [16, 6, 是], [18, 4, 是], [20, 2, 是]], 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]]], 是]

source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

derived = [[[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], 是]

source roles = [0, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]]

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], [0, 12, 12, 0], [0, 14, 14, 0], [0, 16, 16, 0], [0, 18, 18, 0], [0, 20, 20, 0], [2, 0, 2, 0], [2, 2, 4, 0], [2, 4, 6, 0], [2, 6, 8, 0], [2, 8, 10, 0], [2, 10, 12, 0], [2, 12, 14, 0], [2, 14, 16, 0], [2, 16, 18, 0], [2, 18, 20, 0], [2, 20, 0, 0], [4, 0, 4, 0], [4, 2, 6, 0], [4, 4, 8, 0], [4, 6, 10, 0], [4, 8, 12, 0], [4, 10, 14, 0], [4, 12, 16, 0], [4, 14, 18, 0], [4, 16, 20, 0], [4, 18, 0, 0], [4, 20, 2, 0], [6, 0, 6, 0], [6, 2, 8, 0], [6, 4, 10, 0], [6, 6, 12, 0], [6, 8, 14, 0], [6, 10, 16, 0], [6, 12, 18, 0], [6, 14, 20, 0], [6, 16, 0, 0], [6, 18, 2, 0], [6, 20, 4, 0], [8, 0, 8, 0], [8, 2, 10, 0], [8, 4, 12, 0], [8, 6, 14, 0], [8, 8, 16, 0], [8, 10, 18, 0], [8, 12, 20, 0], [8, 14, 0, 0], [8, 16, 2, 0], [8, 18, 4, 0], [8, 20, 6, 0], [10, 0, 10, 0], [10, 2, 12, 0], [10, 4, 14, 0], [10, 6, 16, 0], [10, 8, 18, 0], [10, 10, 20, 0], [10, 12, 0, 0], [10, 14, 2, 0], [10, 16, 4, 0], [10, 18, 6, 0], [10, 20, 8, 0], [12, 0, 12, 0], [12, 2, 14, 0], [12, 4, 16, 0], [12, 6, 18, 0], [12, 8, 20, 0], [12, 10, 0, 0], [12, 12, 2, 0], [12, 14, 4, 0], [12, 16, 6, 0], [12, 18, 8, 0], [12, 20, 10, 0], [14, 0, 14, 0], [14, 2, 16, 0], [14, 4, 18, 0], [14, 6, 20, 0], [14, 8, 0, 0], [14, 10, 2, 0], [14, 12, 4, 0], [14, 14, 6, 0], [14, 16, 8, 0], [14, 18, 10, 0], [14, 20, 12, 0], [16, 0, 16, 0], [16, 2, 18, 0], [16, 4, 20, 0], [16, 6, 0, 0], [16, 8, 2, 0], [16, 10, 4, 0], [16, 12, 6, 0], [16, 14, 8, 0], [16, 16, 10, 0], [16, 18, 12, 0], [16, 20, 14, 0], [18, 0, 18, 0], [18, 2, 20, 0], [18, 4, 0, 0], [18, 6, 2, 0], [18, 8, 4, 0], [18, 10, 6, 0], [18, 12, 8, 0], [18, 14, 10, 0], [18, 16, 12, 0], [18, 18, 14, 0], [18, 20, 16, 0], [20, 0, 20, 0], [20, 2, 0, 0], [20, 4, 2, 0], [20, 6, 4, 0], [20, 8, 6, 0], [20, 10, 8, 0], [20, 12, 10, 0], [20, 14, 12, 0], [20, 16, 14, 0], [20, 18, 16, 0], [20, 20, 18, 0]], [0], 是]

source roles = [0, 1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21]]

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], [0, 13, 13, 1], [0, 15, 15, 1], [0, 17, 17, 1], [0, 19, 19, 1], [0, 21, 21, 1], [2, 1, 3, 1], [2, 3, 5, 1], [2, 5, 7, 1], [2, 7, 9, 1], [2, 9, 11, 1], [2, 11, 13, 1], [2, 13, 15, 1], [2, 15, 17, 1], [2, 17, 19, 1], [2, 19, 21, 1], [2, 21, 1, 1], [4, 1, 5, 1], [4, 3, 7, 1], [4, 5, 9, 1], [4, 7, 11, 1], [4, 9, 13, 1], [4, 11, 15, 1], [4, 13, 17, 1], [4, 15, 19, 1], [4, 17, 21, 1], [4, 19, 1, 1], [4, 21, 3, 1], [6, 1, 7, 1], [6, 3, 9, 1], [6, 5, 11, 1], [6, 7, 13, 1], [6, 9, 15, 1], [6, 11, 17, 1], [6, 13, 19, 1], [6, 15, 21, 1], [6, 17, 1, 1], [6, 19, 3, 1], [6, 21, 5, 1], [8, 1, 9, 1], [8, 3, 11, 1], [8, 5, 13, 1], [8, 7, 15, 1], [8, 9, 17, 1], [8, 11, 19, 1], [8, 13, 21, 1], [8, 15, 1, 1], [8, 17, 3, 1], [8, 19, 5, 1], [8, 21, 7, 1], [10, 1, 11, 1], [10, 3, 13, 1], [10, 5, 15, 1], [10, 7, 17, 1], [10, 9, 19, 1], [10, 11, 21, 1], [10, 13, 1, 1], [10, 15, 3, 1], [10, 17, 5, 1], [10, 19, 7, 1], [10, 21, 9, 1], [12, 1, 13, 1], [12, 3, 15, 1], [12, 5, 17, 1], [12, 7, 19, 1], [12, 9, 21, 1], [12, 11, 1, 1], [12, 13, 3, 1], [12, 15, 5, 1], [12, 17, 7, 1], [12, 19, 9, 1], [12, 21, 11, 1], [14, 1, 15, 1], [14, 3, 17, 1], [14, 5, 19, 1], [14, 7, 21, 1], [14, 9, 1, 1], [14, 11, 3, 1], [14, 13, 5, 1], [14, 15, 7, 1], [14, 17, 9, 1], [14, 19, 11, 1], [14, 21, 13, 1], [16, 1, 17, 1], [16, 3, 19, 1], [16, 5, 21, 1], [16, 7, 1, 1], [16, 9, 3, 1], [16, 11, 5, 1], [16, 13, 7, 1], [16, 15, 9, 1], [16, 17, 11, 1], [16, 19, 13, 1], [16, 21, 15, 1], [18, 1, 19, 1], [18, 3, 21, 1], [18, 5, 1, 1], [18, 7, 3, 1], [18, 9, 5, 1], [18, 11, 7, 1], [18, 13, 9, 1], [18, 15, 11, 1], [18, 17, 13, 1], [18, 19, 15, 1], [18, 21, 17, 1], [20, 1, 21, 1], [20, 3, 1, 1], [20, 5, 3, 1], [20, 7, 5, 1], [20, 9, 7, 1], [20, 11, 9, 1], [20, 13, 11, 1], [20, 15, 13, 1], [20, 17, 15, 1], [20, 19, 17, 1], [20, 21, 19, 1]], [1], 是]

source roles = [1, 0, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]]

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], [1, 12, 13, 1], [1, 14, 15, 1], [1, 16, 17, 1], [1, 18, 19, 1], [1, 20, 21, 1], [3, 0, 3, 1], [3, 2, 5, 1], [3, 4, 7, 1], [3, 6, 9, 1], [3, 8, 11, 1], [3, 10, 13, 1], [3, 12, 15, 1], [3, 14, 17, 1], [3, 16, 19, 1], [3, 18, 21, 1], [3, 20, 1, 1], [5, 0, 5, 1], [5, 2, 7, 1], [5, 4, 9, 1], [5, 6, 11, 1], [5, 8, 13, 1], [5, 10, 15, 1], [5, 12, 17, 1], [5, 14, 19, 1], [5, 16, 21, 1], [5, 18, 1, 1], [5, 20, 3, 1], [7, 0, 7, 1], [7, 2, 9, 1], [7, 4, 11, 1], [7, 6, 13, 1], [7, 8, 15, 1], [7, 10, 17, 1], [7, 12, 19, 1], [7, 14, 21, 1], [7, 16, 1, 1], [7, 18, 3, 1], [7, 20, 5, 1], [9, 0, 9, 1], [9, 2, 11, 1], [9, 4, 13, 1], [9, 6, 15, 1], [9, 8, 17, 1], [9, 10, 19, 1], [9, 12, 21, 1], [9, 14, 1, 1], [9, 16, 3, 1], [9, 18, 5, 1], [9, 20, 7, 1], [11, 0, 11, 1], [11, 2, 13, 1], [11, 4, 15, 1], [11, 6, 17, 1], [11, 8, 19, 1], [11, 10, 21, 1], [11, 12, 1, 1], [11, 14, 3, 1], [11, 16, 5, 1], [11, 18, 7, 1], [11, 20, 9, 1], [13, 0, 13, 1], [13, 2, 15, 1], [13, 4, 17, 1], [13, 6, 19, 1], [13, 8, 21, 1], [13, 10, 1, 1], [13, 12, 3, 1], [13, 14, 5, 1], [13, 16, 7, 1], [13, 18, 9, 1], [13, 20, 11, 1], [15, 0, 15, 1], [15, 2, 17, 1], [15, 4, 19, 1], [15, 6, 21, 1], [15, 8, 1, 1], [15, 10, 3, 1], [15, 12, 5, 1], [15, 14, 7, 1], [15, 16, 9, 1], [15, 18, 11, 1], [15, 20, 13, 1], [17, 0, 17, 1], [17, 2, 19, 1], [17, 4, 21, 1], [17, 6, 1, 1], [17, 8, 3, 1], [17, 10, 5, 1], [17, 12, 7, 1], [17, 14, 9, 1], [17, 16, 11, 1], [17, 18, 13, 1], [17, 20, 15, 1], [19, 0, 19, 1], [19, 2, 21, 1], [19, 4, 1, 1], [19, 6, 3, 1], [19, 8, 5, 1], [19, 10, 7, 1], [19, 12, 9, 1], [19, 14, 11, 1], [19, 16, 13, 1], [19, 18, 15, 1], [19, 20, 17, 1], [21, 0, 21, 1], [21, 2, 1, 1], [21, 4, 3, 1], [21, 6, 5, 1], [21, 8, 7, 1], [21, 10, 9, 1], [21, 12, 11, 1], [21, 14, 13, 1], [21, 16, 15, 1], [21, 18, 17, 1], [21, 20, 19, 1]], [1], 是]

source roles = [1, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21]]

derived = [[[1, 1, 2, 0], [1, 3, 4, 0], [1, 5, 6, 0], [1, 7, 8, 0], [1, 9, 10, 0], [1, 11, 12, 0], [1, 13, 14, 0], [1, 15, 16, 0], [1, 17, 18, 0], [1, 19, 20, 0], [1, 21, 0, 0], [3, 1, 4, 0], [3, 3, 6, 0], [3, 5, 8, 0], [3, 7, 10, 0], [3, 9, 12, 0], [3, 11, 14, 0], [3, 13, 16, 0], [3, 15, 18, 0], [3, 17, 20, 0], [3, 19, 0, 0], [3, 21, 2, 0], [5, 1, 6, 0], [5, 3, 8, 0], [5, 5, 10, 0], [5, 7, 12, 0], [5, 9, 14, 0], [5, 11, 16, 0], [5, 13, 18, 0], [5, 15, 20, 0], [5, 17, 0, 0], [5, 19, 2, 0], [5, 21, 4, 0], [7, 1, 8, 0], [7, 3, 10, 0], [7, 5, 12, 0], [7, 7, 14, 0], [7, 9, 16, 0], [7, 11, 18, 0], [7, 13, 20, 0], [7, 15, 0, 0], [7, 17, 2, 0], [7, 19, 4, 0], [7, 21, 6, 0], [9, 1, 10, 0], [9, 3, 12, 0], [9, 5, 14, 0], [9, 7, 16, 0], [9, 9, 18, 0], [9, 11, 20, 0], [9, 13, 0, 0], [9, 15, 2, 0], [9, 17, 4, 0], [9, 19, 6, 0], [9, 21, 8, 0], [11, 1, 12, 0], [11, 3, 14, 0], [11, 5, 16, 0], [11, 7, 18, 0], [11, 9, 20, 0], [11, 11, 0, 0], [11, 13, 2, 0], [11, 15, 4, 0], [11, 17, 6, 0], [11, 19, 8, 0], [11, 21, 10, 0], [13, 1, 14, 0], [13, 3, 16, 0], [13, 5, 18, 0], [13, 7, 20, 0], [13, 9, 0, 0], [13, 11, 2, 0], [13, 13, 4, 0], [13, 15, 6, 0], [13, 17, 8, 0], [13, 19, 10, 0], [13, 21, 12, 0], [15, 1, 16, 0], [15, 3, 18, 0], [15, 5, 20, 0], [15, 7, 0, 0], [15, 9, 2, 0], [15, 11, 4, 0], [15, 13, 6, 0], [15, 15, 8, 0], [15, 17, 10, 0], [15, 19, 12, 0], [15, 21, 14, 0], [17, 1, 18, 0], [17, 3, 20, 0], [17, 5, 0, 0], [17, 7, 2, 0], [17, 9, 4, 0], [17, 11, 6, 0], [17, 13, 8, 0], [17, 15, 10, 0], [17, 17, 12, 0], [17, 19, 14, 0], [17, 21, 16, 0], [19, 1, 20, 0], [19, 3, 0, 0], [19, 5, 2, 0], [19, 7, 4, 0], [19, 9, 6, 0], [19, 11, 8, 0], [19, 13, 10, 0], [19, 15, 12, 0], [19, 17, 14, 0], [19, 19, 16, 0], [19, 21, 18, 0], [21, 1, 0, 0], [21, 3, 2, 0], [21, 5, 4, 0], [21, 7, 6, 0], [21, 9, 8, 0], [21, 11, 10, 0], [21, 13, 12, 0], [21, 15, 14, 0], [21, 17, 16, 0], [21, 19, 18, 0], [21, 21, 20, 0]], [0], 是]

source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], 是, [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21]]]

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, 12, 14, 16, 18, 20], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21]], [[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], 是, 否, coset_product_preserves_identity_and_inverse_but_need_not_be_commutative)

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。