# 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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r01-finite-group-action-0100/)

Finite group action · qf-mod38-groupaction-nonproperty-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r01-finite-group-action-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261107-r01-finite-group-action-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r01-finite-group-action-foundations/validation-report.json)