# Complete Point Stabilizers, Subgroup Closure and Membership Boundaries

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-20261109-r01-finite-group-stabilizer-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. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(0, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (0, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0001/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 2. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(1, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (1, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0002/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 3. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0003/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 4. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(3, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (3, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0004/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 5. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(4, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (4, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0005/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 6. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0, 1], 2, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0, 1], 2, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0006/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 7. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0007/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 8. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0008/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 9. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0009/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 10. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0010/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 11. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0, 1], 2, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0, 1], 2, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0011/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 12. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(1, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (1, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0012/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 13. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(0, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (0, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0013/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 14. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(6, [0, 1, 2, 3, 4], 5, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4], 5, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0014/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 15. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(6, [0, 1, 2, 3, 4, 5], 6, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 5], 6, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0015/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 16. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(0, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (0, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0016/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 17. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(4, [0, 1, 2], 3, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (4, [0, 1, 2], 3, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0017/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 18. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(3, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (3, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0018/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 19. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0019/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 20. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(1, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (1, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0020/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 21. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(2, [0, 1], 2, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (2, [0, 1], 2, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0021/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 22. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(0, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (0, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0022/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 23. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(6, [0, 1, 2, 3], 4, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (6, [0, 1, 2, 3], 4, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0023/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 24. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=5. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(5, [0, 1, 2, 3, 4], 5, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4], 5, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0024/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 25. The Point Stabilizer Enumerates Every Fixing Group Element

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (point, stabilizer, stabilizer_order, full_enumeration_certificate, certificate).

View answer and explanation(4, [0], 1, Yes, all_group_elements_checked)

Enumerate every g with g*x=x and retain them in group-index order. The result is a subgroup of G, not the orbit in the point carrier or merely one fixed-element witness. The exact result, retaining the requested order, is (4, [0], 1, Yes, all_group_elements_checked).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0025/)

Stabilizer of a finite group action · qf-mod40-stabilizer-stabilizer-of-point · candidate





## 26. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0026/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 27. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0027/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 28. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0028/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 29. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0029/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 30. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0030/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 31. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0031/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 32. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0032/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 33. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0033/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 34. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0034/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 35. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0035/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 36. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0036/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 37. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0037/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 38. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0038/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 39. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 1, 2, 3, 4], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0039/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 40. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0040/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 41. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0041/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 42. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 1, 2], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0042/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 43. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0043/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 44. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0044/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 45. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0045/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 46. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0046/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 47. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0047/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 48. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 1, 2, 3], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2, 3], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0048/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 49. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=5. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0, 1, 2, 3, 4], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0049/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 50. Identity, Product and Inverse Closure Certify the Full Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Give (stabilizer, identity, contains_identity, product_closed, inverse_closed, subgroup, certificate).

View answer and explanation([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure)

First compute the entire stabilizer, then separately check identity membership, every product of its members and the inverse of every member using the same group table. All three obligations supply the subgroup flag. The exact result, retaining the requested order, is ([0], 0, Yes, Yes, Yes, Yes, identity_product_inverse_closure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0050/)

Stabilizer of a finite group action · qf-mod40-stabilizer-subgroup-certificate · candidate





## 51. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(0, 0, 0, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 0, 0, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0051/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 52. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(1, 1, 2, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 1, 2, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0052/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 53. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(2, 2, 0, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (2, 2, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0053/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 54. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(3, 3, 1, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 3, 1, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0054/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 55. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Test group_element=4. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(4, 4, 2, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (4, 4, 2, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0055/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 56. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(1, 2, 2, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 2, 2, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0056/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 57. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(0, 2, 2, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 2, 2, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0057/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 58. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(3, 2, 1, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 2, 1, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0058/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 59. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(3, 2, 0, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 2, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0059/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 60. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(3, 2, 5, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 2, 5, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0060/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 61. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(0, 2, 2, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 2, 2, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0061/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 62. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(2, 1, 0, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (2, 1, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0062/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 63. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(0, 0, 0, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 0, 0, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0063/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 64. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(3, 6, 6, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 6, 6, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0064/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 65. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(2, 6, 6, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (2, 6, 6, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0065/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 66. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(1, 0, 1, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 0, 1, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0066/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 67. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(1, 4, 4, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 4, 4, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0067/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 68. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(1, 3, 0, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 3, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0068/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 69. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(3, 2, 0, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 2, 0, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0069/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 70. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Test group_element=1. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(1, 1, 2, No, No)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (1, 1, 2, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0070/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 71. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(0, 2, 2, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 2, 2, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0071/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 72. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(0, 0, 0, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 0, 0, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0072/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 73. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(2, 6, 6, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (2, 6, 6, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0073/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 74. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=5. Test group_element=3. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(3, 5, 5, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (3, 5, 5, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0074/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 75. Fixing This Point Is Exactly Membership in Its Stabilizer

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Test group_element=0. Give (group_element, point, image, fixes_point, in_stabilizer).

View answer and explanation(0, 4, 4, Yes, Yes)

Read its action image, compare with the specified point and separately report membership in the complete stabilizer. An image different from x makes both flags No. The exact result, retaining the requested order, is (0, 4, 4, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0075/)

Stabilizer of a finite group action · qf-mod40-stabilizer-membership-audit · candidate





## 76. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Compare other_point=1. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0076/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 77. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Compare other_point=2. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0077/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 78. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0078/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 79. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Compare other_point=4. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(3, [0], 4, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (3, [0], 4, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0079/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 80. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Compare other_point=5. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(4, [0], 5, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (4, [0], 5, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0080/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 81. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=0. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0, 1], 0, [0], No, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0, 1], 0, [0], No, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0081/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 82. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0], 3, [0, 1, 2], No, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0, 1, 2], No, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0082/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 83. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0083/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 84. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0084/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 85. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0085/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 86. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0, 1], 3, [0, 1], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0, 1], 3, [0, 1], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0086/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 87. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Compare other_point=2. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0087/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 88. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Compare other_point=1. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0088/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 89. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Compare other_point=0. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(6, [0, 1, 2, 3, 4], 0, [0], No, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4], 0, [0], No, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0089/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 90. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Compare other_point=7. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(6, [0, 1, 2, 3, 4, 5], 7, [0, 1, 2, 3, 4, 5], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 5], 7, [0, 1, 2, 3, 4, 5], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0090/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 91. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Compare other_point=1. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0091/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 92. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Compare other_point=5. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(4, [0, 1, 2], 5, [0, 1, 2], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (4, [0, 1, 2], 5, [0, 1, 2], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0092/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 93. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=3. Compare other_point=4. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(3, [0], 4, [0, 1, 2, 3], No, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (3, [0], 4, [0, 1, 2, 3], No, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0093/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 94. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0], 3, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0094/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 95. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=1. Compare other_point=2. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (1, [0], 2, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0095/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 96. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1], [1, 0]], identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=2. Compare other_point=3. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(2, [0, 1], 3, [0, 1], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (2, [0, 1], 3, [0, 1], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0096/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 97. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=0. Compare other_point=1. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (0, [0], 1, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0097/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 98. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Compare other_point=7. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(6, [0, 1, 2, 3], 7, [0, 1, 2, 3], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (6, [0, 1, 2, 3], 7, [0, 1, 2, 3], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0098/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 99. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group 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]], 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=5. Compare other_point=6. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(5, [0, 1, 2, 3, 4], 6, [0, 1, 2, 3, 4], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4], 6, [0, 1, 2, 3, 4], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0099/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate





## 100. Point Stabilizers May Agree without Becoming Orbits or Kernels

Use complete group table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, 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]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=4. Compare other_point=5. Give (point, stabilizer, other_point, other_stabilizer, same_stabilizer, orbit_or_kernel_identification_claim, boundary).

The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets.

View answer and explanation(4, [0], 5, [0], Yes, No, stabilizer_not_orbit_or_action_kernel)

Compute both complete stabilizers and their equality flag. The final fixed No withholds identification of stabilizer, orbit and action kernel as concepts; it does not negate equal stabilizers or rule out special set equalities. The exact result, retaining the requested order, is (4, [0], 5, [0], Yes, No, stabilizer_not_orbit_or_action_kernel).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r01-finite-group-stabilizer-0100/)

Stabilizer of a finite group action · qf-mod40-stabilizer-point-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261109-r01-finite-group-stabilizer-foundations/validation-report.json)