有限正規子群基礎 · 脈衝 R01
本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。
本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。
batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations · 2026-10-15 · 100 題
欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。
欄序與來源綁定資料
1. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"subset":[0]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
2. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"subset":[0,1,2,3,4,5]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
3. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_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]],"identity":0,"subset":[0,2,4,6]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
4. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[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],[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]],"identity":0,"subset":[0,5]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
5. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；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":[0,1]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 1], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 1], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
6. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"identity":1,"subset":[1]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([1], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([1], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
7. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],"identity":1,"subset":[0,1,2,3,4,5]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
8. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[7,0,1,2,3,4,5,6],[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]],"identity":1,"subset":[1,3,5,7]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([1, 3, 5, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([1, 3, 5, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
9. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[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],[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]],"identity":1,"subset":[1,6]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([1, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([1, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
10. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[11,0,1,2,3,4,5,6,7,8,9,10],[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]],"identity":1,"subset":[1,2]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([1, 2], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)
Wave6第七層核對=([1, 2], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
11. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"identity":2,"subset":[2]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([2], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([2], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
12. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],"identity":2,"subset":[0,1,2,3,4,5]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
13. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[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]],"identity":2,"subset":[0,2,4,6]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
14. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[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],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],"identity":2,"subset":[2,7]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([2, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([2, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
15. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":2,"subset":[2,3]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([2, 3], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)
Wave6第七層核對=([2, 3], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
16. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"identity":3,"subset":[3]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([3], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([3], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
17. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],"identity":3,"subset":[0,1,2,3,4,5]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
18. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[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]],"identity":3,"subset":[1,3,5,7]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([1, 3, 5, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([1, 3, 5, 7], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
19. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[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],[6,7,8,9,0,1,2,3,4,5]],"identity":3,"subset":[3,8]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([3, 8], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([3, 8], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
20. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[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],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":3,"subset":[3,4]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([3, 4], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)
Wave6第七層核對=([3, 4], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
21. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"identity":0,"subset":[0]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
22. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],"identity":4,"subset":[0,1,2,3,4,5]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
23. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[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],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],"identity":4,"subset":[0,2,4,6]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([0, 2, 4, 6], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
24. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；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":[4,9]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([4, 9], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)
Wave6第七層核對=([4, 9], 是, 是, 是, 是, 是, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
25. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-subgroup-qualification；資料={"group_table":[[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],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":4,"subset":[4,5]}。請依 [subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate] 欄序回答。
查看答案與解釋
([4, 5], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)
Wave6第七層核對=([4, 5], 是, 否, 否, 否, 否, subgroup_obligations_precede_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-subgroup-qualification · candidate
26. 有限正規子群
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":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"subset":[0]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
27. 有限正規子群
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":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"subset":[0,1,2,3,4,5]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
28. 有限正規子群
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":[[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]],"identity":0,"subset":[0]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0], [[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, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0], [[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, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
29. 有限正規子群
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":[[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],[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]],"identity":0,"subset":[0,1,2,3,4,5,6,7,8,9]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([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)
Wave6第七層核對=([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)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
30. 有限正規子群
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":[[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]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是], [6, [0], 是], [7, [0], 是], [8, [0], 是], [9, [0], 是], [10, [0], 是], [11, [0], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
31. 有限正規子群
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":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"identity":1,"subset":[0,1,2,3]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3], [[0, [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3], [[0, [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
32. 有限正規子群
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":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],"identity":1,"subset":[1]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([1], [[0, [1], 是], [1, [1], 是], [2, [1], 是], [3, [1], 是], [4, [1], 是], [5, [1], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([1], [[0, [1], 是], [1, [1], 是], [2, [1], 是], [3, [1], 是], [4, [1], 是], [5, [1], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
33. 有限正規子群
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":[[7,0,1,2,3,4,5,6],[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]],"identity":1,"subset":[0,1,2,3,4,5,6,7]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
34. 有限正規子群
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":[[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],[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]],"identity":1,"subset":[1]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([1], [[0, [1], 是], [1, [1], 是], [2, [1], 是], [3, [1], 是], [4, [1], 是], [5, [1], 是], [6, [1], 是], [7, [1], 是], [8, [1], 是], [9, [1], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([1], [[0, [1], 是], [1, [1], 是], [2, [1], 是], [3, [1], 是], [4, [1], 是], [5, [1], 是], [6, [1], 是], [7, [1], 是], [8, [1], 是], [9, [1], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
35. 有限正規子群
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":[[11,0,1,2,3,4,5,6,7,8,9,10],[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]],"identity":1,"subset":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
36. 有限正規子群
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":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"identity":2,"subset":[2]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
37. 有限正規子群
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":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],"identity":2,"subset":[0,1,2,3,4,5]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
38. 有限正規子群
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":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[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]],"identity":2,"subset":[2]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是], [4, [2], 是], [5, [2], 是], [6, [2], 是], [7, [2], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是], [4, [2], 是], [5, [2], 是], [6, [2], 是], [7, [2], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
39. 有限正規子群
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":[[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],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],"identity":2,"subset":[0,1,2,3,4,5,6,7,8,9]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([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)
Wave6第七層核對=([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)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
40. 有限正規子群
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":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":2,"subset":[2]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是], [4, [2], 是], [5, [2], 是], [6, [2], 是], [7, [2], 是], [8, [2], 是], [9, [2], 是], [10, [2], 是], [11, [2], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([2], [[0, [2], 是], [1, [2], 是], [2, [2], 是], [3, [2], 是], [4, [2], 是], [5, [2], 是], [6, [2], 是], [7, [2], 是], [8, [2], 是], [9, [2], 是], [10, [2], 是], [11, [2], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
41. 有限正規子群
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":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"identity":3,"subset":[0,1,2,3]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3], [[0, [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3], [[0, [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
42. 有限正規子群
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":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],"identity":3,"subset":[3]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([3], [[0, [3], 是], [1, [3], 是], [2, [3], 是], [3, [3], 是], [4, [3], 是], [5, [3], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([3], [[0, [3], 是], [1, [3], 是], [2, [3], 是], [3, [3], 是], [4, [3], 是], [5, [3], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
43. 有限正規子群
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":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[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]],"identity":3,"subset":[0,1,2,3,4,5,6,7]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
44. 有限正規子群
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":[[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],[6,7,8,9,0,1,2,3,4,5]],"identity":3,"subset":[3]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([3], [[0, [3], 是], [1, [3], 是], [2, [3], 是], [3, [3], 是], [4, [3], 是], [5, [3], 是], [6, [3], 是], [7, [3], 是], [8, [3], 是], [9, [3], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([3], [[0, [3], 是], [1, [3], 是], [2, [3], 是], [3, [3], 是], [4, [3], 是], [5, [3], 是], [6, [3], 是], [7, [3], 是], [8, [3], 是], [9, [3], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
45. 有限正規子群
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":[[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],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":3,"subset":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
46. 有限正規子群
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":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"identity":0,"subset":[0]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0], [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
47. 有限正規子群
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":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],"identity":4,"subset":[0,1,2,3,4,5]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
48. 有限正規子群
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":[[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],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],"identity":4,"subset":[4]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([4], [[0, [4], 是], [1, [4], 是], [2, [4], 是], [3, [4], 是], [4, [4], 是], [5, [4], 是], [6, [4], 是], [7, [4], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([4], [[0, [4], 是], [1, [4], 是], [2, [4], 是], [3, [4], 是], [4, [4], 是], [5, [4], 是], [6, [4], 是], [7, [4], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
49. 有限正規子群
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 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([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)
Wave6第七層核對=([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)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
50. 有限正規子群
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":[[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],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":4,"subset":[4]}。請依 [subgroup, conjugate_rows, normal, certificate] 欄序回答。
查看答案與解釋
([4], [[0, [4], 是], [1, [4], 是], [2, [4], 是], [3, [4], 是], [4, [4], 是], [5, [4], 是], [6, [4], 是], [7, [4], 是], [8, [4], 是], [9, [4], 是], [10, [4], 是], [11, [4], 是]], 是, every_group_element_conjugates_H_to_itself)
Wave6第七層核對=([4], [[0, [4], 是], [1, [4], 是], [2, [4], 是], [3, [4], 是], [4, [4], 是], [5, [4], 是], [6, [4], 是], [7, [4], 是], [8, [4], 是], [9, [4], 是], [10, [4], 是], [11, [4], 是]], 是, every_group_element_conjugates_H_to_itself)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-conjugate-set-audit · candidate
51. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[0,1,2,3],[1,2,3,0],[2,3,0,1],[3,0,1,2]],"identity":0,"subset":[0]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
52. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],"identity":0,"subset":[0,1,2,3,4,5]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
53. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_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]],"identity":0,"subset":[0]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
54. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[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],[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]],"identity":0,"subset":[0,1,2,3,4,5,6,7,8,9]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
55. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；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]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
56. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[3,0,1,2],[0,1,2,3],[1,2,3,0],[2,3,0,1]],"identity":1,"subset":[0,1,2,3]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3], [[0, [0, 1, 2, 3], [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3], [[0, [0, 1, 2, 3], [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
57. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3]],"identity":1,"subset":[1]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([1], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([1], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
58. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[7,0,1,2,3,4,5,6],[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]],"identity":1,"subset":[0,1,2,3,4,5,6,7]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
59. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[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],[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]],"identity":1,"subset":[1]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([1], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([1], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
60. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[11,0,1,2,3,4,5,6,7,8,9,10],[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]],"identity":1,"subset":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
61. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[2,3,0,1],[3,0,1,2],[0,1,2,3],[1,2,3,0]],"identity":2,"subset":[2]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([2], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([2], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
62. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2]],"identity":2,"subset":[0,1,2,3,4,5]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
63. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[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]],"identity":2,"subset":[2]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([2], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([2], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
64. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[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],[6,7,8,9,0,1,2,3,4,5],[7,8,9,0,1,2,3,4,5,6]],"identity":2,"subset":[0,1,2,3,4,5,6,7,8,9]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
65. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":2,"subset":[2]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([2], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([2], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
66. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],"identity":3,"subset":[0,1,2,3]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3], [[0, [0, 1, 2, 3], [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3], [[0, [0, 1, 2, 3], [0, 1, 2, 3], 是], [1, [0, 1, 2, 3], [0, 1, 2, 3], 是], [2, [0, 1, 2, 3], [0, 1, 2, 3], 是], [3, [0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
67. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1]],"identity":3,"subset":[3]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([3], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([3], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是], [4, [4], [4], 是], [5, [5], [5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
68. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[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]],"identity":3,"subset":[0,1,2,3,4,5,6,7]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
69. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[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],[6,7,8,9,0,1,2,3,4,5]],"identity":3,"subset":[3]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([3], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([3], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
70. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[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],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":3,"subset":[0,1,2,3,4,5,6,7,8,9,10,11]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是], [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
71. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[0,1,2,3],[1,0,3,2],[2,3,1,0],[3,2,0,1]],"identity":0,"subset":[0]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0], [[0, [0], [0], 是], [1, [1], [1], 是], [2, [2], [2], 是], [3, [3], [3], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
72. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4],[0,1,2,3,4,5],[1,2,3,4,5,0]],"identity":4,"subset":[0,1,2,3,4,5]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5], [[0, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [2, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [3, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [4, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是], [5, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
73. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[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],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2]],"identity":4,"subset":[4]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([4], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([4], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
74. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-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 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [[0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是], [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
75. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={"group_table":[[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],[0,1,2,3,4,5,6,7,8,9,10,11],[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]],"identity":4,"subset":[4]}。請依 [subgroup, coset_rows, all_left_right_equal, certificate] 欄序回答。
查看答案與解釋
([4], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)
Wave6第七層核對=([4], [[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]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-left-right-coset-audit · candidate
76. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,0,5,1,3],[3,5,4,0,2,1],[4,2,3,1,5,0],[5,3,1,2,0,4]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 4, 1, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 4, 1, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
77. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 5, 1, 4, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 5, 1, 4, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
78. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,0,1,5,4],[3,2,4,5,1,0],[4,5,3,2,0,1],[5,4,1,0,2,3]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 3, 1, 5, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 3, 1, 5, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
79. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,0,4,3,1],[3,4,1,5,2,0],[4,3,5,1,0,2],[5,2,4,0,1,3]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 5, 1, 3, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 5, 1, 3, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
80. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,0,1,5,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,4,3,2,1,0]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 3, 1, 4, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 3, 1, 4, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
81. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,0,5,1,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,3,4,1,2,0]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 4, 1, 3, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 4, 1, 3, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
82. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 4, 1, 5, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 4, 1, 5, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
83. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,0,5,1,3],[3,5,4,0,2,1],[4,2,3,1,5,0],[5,3,1,2,0,4]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 5, 1, 4, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 5, 1, 4, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
84. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,5,4,1,0],[3,2,1,0,5,4],[4,5,3,2,0,1],[5,4,0,1,3,2]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 2, 1, 5, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 2, 1, 5, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
85. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,5,1,3,0],[3,5,4,0,2,1],[4,2,1,5,0,3],[5,3,0,4,1,2]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 5, 1, 2, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 5, 1, 2, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
86. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,4,5,0,1],[3,2,1,0,5,4],[4,5,0,1,2,3],[5,4,3,2,1,0]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 2, 1, 4, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 2, 1, 4, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
87. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,4,1,0,3],[3,4,5,0,1,2],[4,3,0,5,2,1],[5,2,1,4,3,0]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 4, 1, 2, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 4, 1, 2, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
88. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,0,4,3,1],[3,4,1,5,2,0],[4,3,5,1,0,2],[5,2,4,0,1,3]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 3, 1, 5, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 3, 1, 5, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
89. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,0,1,5,4],[3,2,4,5,1,0],[4,5,3,2,0,1],[5,4,1,0,2,3]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 5, 1, 3, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 5, 1, 3, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
90. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,5,1,3,0],[3,5,4,0,2,1],[4,2,1,5,0,3],[5,3,0,4,1,2]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 2, 1, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 2, 1, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
91. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,5,4,1,0],[3,2,1,0,5,4],[4,5,3,2,0,1],[5,4,0,1,3,2]],"identity":0,"subset":[0,1],"witness_g":5,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 5, 1, 2, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 5, 1, 2, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
92. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,2,1,5,0,3],[5,3,4,1,2,0]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 2, 1, 3, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 2, 1, 3, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
93. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,3,5,1,0,2],[5,2,1,4,3,0]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 3, 1, 2, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 3, 1, 2, 5, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
94. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,4,0,5,1,3],[3,5,1,4,0,2],[4,2,5,0,3,1],[5,3,4,1,2,0]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 3, 1, 4, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 3, 1, 4, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
95. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,3,0,1,5,4],[3,2,5,4,0,1],[4,5,1,0,3,2],[5,4,3,2,1,0]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 4, 1, 3, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 4, 1, 3, 2, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
96. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,3,2,5,4],[2,5,4,1,0,3],[3,4,5,0,1,2],[4,3,0,5,2,1],[5,2,1,4,3,0]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 2, 1, 4, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 2, 1, 4, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
97. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,3,4,5,0,1],[3,2,1,0,5,4],[4,5,0,1,2,3],[5,4,3,2,1,0]],"identity":0,"subset":[0,1],"witness_g":4,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 4, 1, 2, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 4, 1, 2, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
98. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,3,5,1,0,2],[5,2,1,4,3,0]],"identity":0,"subset":[0,1],"witness_g":2,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 2, 1, 3, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 2, 1, 3, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
99. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,5,4,3,2],[2,4,3,0,5,1],[3,5,0,2,1,4],[4,2,1,5,0,3],[5,3,4,1,2,0]],"identity":0,"subset":[0,1],"witness_g":3,"witness_h":1}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 1], 3, 1, 2, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 1], 3, 1, 2, 4, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
100. 有限正規子群
完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-nonnormal-witness；資料={"group_table":[[0,1,2,3,4,5],[1,0,4,5,2,3],[2,5,0,4,3,1],[3,4,5,0,1,2],[4,3,1,2,5,0],[5,2,3,1,0,4]],"identity":0,"subset":[0,2],"witness_g":4,"witness_h":2}。請依 [subgroup, g, h, inverse_g, escaping_conjugate, normal, certificate] 欄序回答。
查看答案與解釋
([0, 2], 4, 2, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)
Wave6第七層核對=([0, 2], 4, 2, 5, 3, 否, one_h_in_H_with_g_h_g_inverse_outside_H_disproves_normality)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。
本題逐步詳解（AI 二次覆核）
有限正規子群 · qf-mod44-normal-nonnormal-witness · candidate
原始題目 JSONL · 驗收報告