Complete Group-Action Laws, Identity Rows and Independent Nonclaims
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261107-r01-finite-group-action-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(2)={0,…,1}, and total action_table=[[0, 1], [1, 0]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(2, 2, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 2, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
2. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(3, 3, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 3, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
3. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(4, 4, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 4, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
4. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(5, 5, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 5, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
5. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_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]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(6, 6, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 6, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
6. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 0, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(2, 3, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 3, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
7. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(3, 4, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 4, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
8. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(4, 5, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 5, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
9. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(5, 6, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 6, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
10. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(6, 7, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 7, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
11. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(2, 4, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 4, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
12. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(3, 5, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 5, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
13. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(4, 6, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 6, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
14. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(5, 7, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 7, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
15. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(6, 8, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 8, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
16. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(2, 5, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 5, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
17. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(3, 6, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 6, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
18. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(4, 7, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 7, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
19. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(5, 8, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 8, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
20. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(6, 9, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 9, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
21. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(2, 6, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (2, 6, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
22. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(3, 7, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (3, 7, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
23. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(4, 8, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (4, 8, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
24. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(5, 9, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (5, 9, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
25. A Group Action Needs Valid Group, Identity and Compatibility Together
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(10)={0,…,9}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action).
View answer and explanation
(6, 10, Yes, Yes, Yes, Yes, Yes)
Audit closure, associativity, the declared identity and inverses on the complete group table. Keep identity-action, all-triple compatibility and permutation-row flags separate; only qualified group plus both action axioms gives validity. The exact result, retaining the requested order, is (6, 10, Yes, Yes, Yes, Yes, Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-qualification · candidate
26. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(2)={0,…,1}, and total action_table=[[0, 1], [1, 0]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
27. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
28. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
29. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
30. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_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]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
31. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 0, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
32. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
33. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
34. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
35. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
36. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
37. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
38. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
39. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
40. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
41. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
42. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
43. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
44. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
45. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
46. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
47. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
48. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
49. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
50. Every Point Must Be Fixed by the Declared Identity Row
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(10)={0,…,9}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).
View answer and explanation
(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes], [9, 9, Yes]], Yes)
List every point, its image under the declared identity and the equality flag. This identity-row audit does not by itself establish the group table or the multiplication-compatibility law. The exact result, retaining the requested order, is (0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes], [9, 9, Yes]], Yes).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-identity-audit · candidate
51. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(2)={0,…,1}, and total action_table=[[0, 1], [1, 0]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(8, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (8, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
52. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(27, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (27, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
53. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(64, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (64, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
54. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(125, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (125, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
55. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_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]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(216, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (216, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
56. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 0, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(12, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (12, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
57. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(36, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (36, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
58. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(80, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (80, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
59. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(150, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (150, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
60. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(252, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (252, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
61. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(16, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (16, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
62. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(45, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (45, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
63. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(96, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (96, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
64. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(175, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (175, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
65. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(288, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (288, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
66. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(20, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (20, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
67. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(54, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (54, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
68. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(112, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (112, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
69. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(200, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (200, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
70. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(324, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (324, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
71. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(24, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (24, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
72. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(63, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (63, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
73. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(128, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (128, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
74. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(225, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (225, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
75. All Group-Group-Point Triples Must Satisfy the Same Action Law
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(10)={0,…,9}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (triple_count, first_failure, compatible, certificate).
View answer and explanation
(360, null, Yes, all_g_h_x_checked)
Check every ordered (g,h,x) triple. Report their total count and the first failure in increasing g,h,x order as [g,h,x,left,right], or literal null when none fails. Compatibility alone does not replace the identity-action or group qualification. The exact result, retaining the requested order, is (360, null, Yes, all_g_h_x_checked).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-compatibility-audit · candidate
76. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(2)={0,…,1}, and total action_table=[[0, 1], [1, 0]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
77. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
78. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
79. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
80. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_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]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
81. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(3)={0,…,2}, and total action_table=[[0, 1, 2], [1, 0, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
82. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
83. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
84. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
85. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
86. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
87. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
88. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
89. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
90. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
91. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
92. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
93. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
94. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
95. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
96. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(6)={0,…,5}, and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
97. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
98. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(8)={0,…,7}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
99. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
100. A Valid Action Does Not Automatically Certify Stronger Action Properties
Use complete Cayley 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(10)={0,…,9}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (valid_action, faithful_claim, free_claim, transitive_claim, boundary).
The three No values withhold faithful, free or transitive claims; they do not prove those properties false.
View answer and explanation
(Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive)
Report actual action validity, followed by three literal No nonclaim markers. Faithful, free and transitive need their own definitions and certificates. The No values do not prove those properties false, even if a particular valid action actually has them. The exact result, retaining the requested order, is (Yes, No, No, No, valid_action_does_not_automatically_claim_faithful_free_or_transitive).
Reviewed step-by-step derivation
Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate
Canonical question JSONL · English JSONL · Validation report