# Complete Group Orbits, Equivalence Witnesses and Partition Certificates

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-20261108-r01-finite-group-orbit-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. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1], 2, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0001/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 2. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=1. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (1, [0, 1, 2], 3, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0002/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 3. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3], 4, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0003/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 4. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=3. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0004/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 5. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=4. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0005/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 6. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

View answer and explanation(2, [2], 2, all_group_elements_enumerated)

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [2], 2, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0006/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 7. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2], 3, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0007/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 8. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3], 4, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0008/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 9. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0009/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 10. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0010/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 11. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

View answer and explanation(2, [2], 2, all_group_elements_enumerated)

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [2], 2, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0011/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 12. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=1. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (1, [0, 1, 2], 3, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0012/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 13. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1, 2, 3], 4, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0013/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 14. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=6. Give (point, orbit, group_order, certificate).

View answer and explanation(6, [6], 5, all_group_elements_enumerated)

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (6, [6], 5, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0014/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 15. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=6. Give (point, orbit, group_order, certificate).

View answer and explanation(6, [6], 6, all_group_elements_enumerated)

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (6, [6], 6, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0015/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 16. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1], 2, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0016/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 17. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=4. Give (point, orbit, group_order, certificate).

View answer and explanation(4, [4], 3, all_group_elements_enumerated)

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (4, [4], 3, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0017/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 18. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=3. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (3, [0, 1, 2, 3], 4, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0018/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 19. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4], 5, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0019/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 20. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=1. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0020/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 21. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=2. Give (point, orbit, group_order, certificate).

View answer and explanation(2, [2], 2, all_group_elements_enumerated)

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (2, [2], 2, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0021/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 22. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=0. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (0, [0, 1, 2], 3, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0022/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 23. An Orbit Uses All Group Elements, Not One Action Image

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=6. Give (point, orbit, group_order, certificate).

View answer and explanation(6, [6], 4, all_group_elements_enumerated)

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (6, [6], 4, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0023/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 24. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=5. Give (point, orbit, group_order, certificate).

View answer and explanation(5, [5], 5, all_group_elements_enumerated)

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (5, [5], 5, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0024/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 25. An Orbit Uses All Group Elements, Not One Action Image

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]], declared identity=0, point carrier Fin(10)={0,…,9} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Point=4. Give (point, orbit, group_order, certificate).

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

Enumerate every g*x, remove repeated point images and order the full orbit increasingly. This is not a stabilizer, a sampled time trajectory or an unqualified single-generator reachability calculation. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5], 6, all_group_elements_enumerated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0025/)

Orbit of a finite group action · qf-mod39-orbit-orbit-of-point · candidate





## 26. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1]], [0, 1], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0026/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 27. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2]], [0, 1, 2], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0027/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 28. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3]], [0, 1, 2, 3], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0028/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 29. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0029/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 30. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5]], [0, 1, 2, 3, 4, 5], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0030/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 31. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1], [2]], [0, 1, 2], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0031/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 32. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2], [3]], [0, 1, 2, 3], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0032/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 33. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4]], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0033/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 34. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0034/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 35. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0035/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 36. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1], [2], [3]], [0, 1, 2, 3], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0036/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 37. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2], [3], [4]], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0037/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 38. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0038/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 39. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0039/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 40. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

View answer and explanation([[0, 1, 2, 3, 4, 5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0040/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 41. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1], [2], [3], [4]], [0, 1, 2, 3, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0041/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 42. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0042/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 43. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0043/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 44. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

View answer and explanation([[0, 1, 2, 3, 4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0044/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 45. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

View answer and explanation([[0, 1, 2, 3, 4, 5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, Yes)

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0045/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 46. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0046/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 47. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

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

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2], [3], [4], [5], [6]], [0, 1, 2, 3, 4, 5, 6], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0047/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 48. The Complete Orbits Partition the Whole Point Carrier

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

View answer and explanation([[0, 1, 2, 3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes)

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4], [5], [6], [7]], [0, 1, 2, 3, 4, 5, 6, 7], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0048/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 49. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

View answer and explanation([[0, 1, 2, 3, 4], [5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, Yes)

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5], [6], [7], [8]], [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0049/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 50. The Complete Orbits Partition the Whole Point Carrier

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]], declared identity=0, point carrier Fin(10)={0,…,9} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate).

View answer and explanation([[0, 1, 2, 3, 4, 5], [6], [7], [8], [9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes, Yes)

Compute one orbit for each point not already covered, order each orbit and order the orbit list by its least point. Retain the complete flattened coverage and the separate pairwise-disjoint flag; the qualified action supplies an equivalence-class partition. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6], [7], [8], [9]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0050/)

Orbit of a finite group action · qf-mod39-orbit-orbit-partition · candidate





## 51. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=0, y=1. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (0, 1, [1], Yes, [0, 1], [0, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0051/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 52. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=1, y=2. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (1, 2, [1], Yes, [0, 1, 2], [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0052/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 53. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0053/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 54. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=3, y=4. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (3, 4, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0054/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 55. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=4, y=5. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (4, 5, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0055/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 56. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=0. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 0, [], No, [2], [0, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0056/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 57. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [], No, [0, 1, 2], [3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0057/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 58. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0058/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 59. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0059/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 60. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0060/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 61. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [], No, [2], [3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0061/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 62. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=1, y=2. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (1, 2, [1], Yes, [0, 1, 2], [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0062/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 63. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=0, y=1. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (0, 1, [1], Yes, [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0063/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 64. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=6, y=0. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (6, 0, [], No, [6], [0, 1, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0064/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 65. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=6, y=7. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

View answer and explanation(6, 7, [], No, [6], [7], Yes)

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (6, 7, [], No, [6], [7], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0065/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 66. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=0, y=1. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (0, 1, [1], Yes, [0, 1], [0, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0066/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 67. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=4, y=5. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (4, 5, [], No, [4], [5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0067/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 68. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=3, y=4. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (3, 4, [], No, [0, 1, 2, 3], [4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0068/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 69. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [1], Yes, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0069/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 70. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=1, y=2. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (1, 2, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0070/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 71. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=2, y=3. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (2, 3, [], No, [2], [3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0071/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 72. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=0, y=1. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (0, 1, [1], Yes, [0, 1, 2], [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0072/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 73. Existence of a Sending Group Element Agrees with Orbit Equality

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=6, y=7. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

View answer and explanation(6, 7, [], No, [6], [7], Yes)

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (6, 7, [], No, [6], [7], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0073/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 74. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=5, y=6. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

View answer and explanation(5, 6, [], No, [5], [6], Yes)

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (5, 6, [], No, [5], [6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0074/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 75. Existence of a Sending Group Element Agrees with Orbit Equality

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]], declared identity=0, point carrier Fin(10)={0,…,9} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Compare x=4, y=5. Give (x, y, sending_elements, same_orbit_relation, orbit_x, orbit_y, relation_iff_orbit_equality).

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

List every group element sending x to y in group-index order. Keep that relation flag distinct from the final equivalence-of-tests flag: when x and y have different orbits, the relation is No while agreement of the two tests is still Yes. The exact result, retaining the requested order, is (4, 5, [1], Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0075/)

Orbit of a finite group action · qf-mod39-orbit-equivalence-audit · candidate





## 76. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(2)={0,…,1} and total action_table=[[0, 1], [1, 0]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=0. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 0, 0, [0, 1], 2, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0076/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 77. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=1. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 1, 2, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0077/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 78. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=2, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (2, 2, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0078/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 79. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=3. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 3, 1, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0079/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 80. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=4, point=4. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (4, 4, 2, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0080/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 81. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0081/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 82. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 2, 0, 3], [2, 0, 1, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 2, 2, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0082/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 83. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 3, 0, 4], [2, 3, 0, 1, 4], [3, 0, 1, 2, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 2, 1, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0083/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 84. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 0, 5], [2, 3, 4, 0, 1, 5], [3, 4, 0, 1, 2, 5], [4, 0, 1, 2, 3, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 2, 0, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0084/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 85. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 0, 6], [2, 3, 4, 5, 0, 1, 6], [3, 4, 5, 0, 1, 2, 6], [4, 5, 0, 1, 2, 3, 6], [5, 0, 1, 2, 3, 4, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 2, 5, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0085/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 86. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(4)={0,…,3} and total action_table=[[0, 1, 2, 3], [1, 0, 2, 3]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0086/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 87. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 2, 0, 3, 4], [2, 0, 1, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=2, point=1. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (2, 1, 0, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0087/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 88. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=0. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 0, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0088/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 89. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 0, 5, 6], [2, 3, 4, 0, 1, 5, 6], [3, 4, 0, 1, 2, 5, 6], [4, 0, 1, 2, 3, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=6. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

View answer and explanation(3, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0089/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 90. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 0, 6, 7], [2, 3, 4, 5, 0, 1, 6, 7], [3, 4, 5, 0, 1, 2, 6, 7], [4, 5, 0, 1, 2, 3, 6, 7], [5, 0, 1, 2, 3, 4, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=2, point=6. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

View answer and explanation(2, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (2, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0090/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 91. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(5)={0,…,4} and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=0. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 0, 1, [0, 1], 2, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0091/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 92. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 3, 4, 5], [2, 0, 1, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=4. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

View answer and explanation(1, 4, 4, [4], 1, No, single_action_image_not_the_whole_orbit)

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 4, 4, [4], 1, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0092/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 93. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 0, 4, 5, 6], [2, 3, 0, 1, 4, 5, 6], [3, 0, 1, 2, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=3. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 3, 0, [0, 1, 2, 3], 4, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0093/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 94. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 2, 0, [0, 1, 2, 3, 4], 5, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0094/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 95. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=1, point=1. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (1, 1, 2, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0095/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 96. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(6)={0,…,5} and total action_table=[[0, 1, 2, 3, 4, 5], [1, 0, 2, 3, 4, 5]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=2. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 2, 2, [2], 1, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0096/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 97. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], declared identity=0, point carrier Fin(7)={0,…,6} and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=0. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 0, 0, [0, 1, 2], 3, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0097/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 98. One Image Alone Is Not a Whole-Orbit Certificate

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], declared identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=2, point=6. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

View answer and explanation(2, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit)

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (2, 6, 6, [6], 1, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0098/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 99. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(9)={0,…,8} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=3, point=5. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

View answer and explanation(3, 5, 5, [5], 1, No, single_action_image_not_the_whole_orbit)

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (3, 5, 5, [5], 1, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0099/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate





## 100. One Image Alone Is Not a Whole-Orbit Certificate

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]], declared identity=0, point carrier Fin(10)={0,…,9} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 0, 6, 7, 8, 9], [2, 3, 4, 5, 0, 1, 6, 7, 8, 9], [3, 4, 5, 0, 1, 2, 6, 7, 8, 9], [4, 5, 0, 1, 2, 3, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 6, 7, 8, 9]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Given group element=0, point=4. Give (group_element, point, image, orbit, orbit_size, single_image_alone_claim, boundary).

The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton.

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

Read this one image separately from the full orbit and its size. No withholds a full-orbit inference from a single image; it does not say that a fully computed orbit cannot be a singleton. The exact result, retaining the requested order, is (0, 4, 4, [0, 1, 2, 3, 4, 5], 6, No, single_action_image_not_the_whole_orbit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0100/)

Orbit of a finite group action · qf-mod39-orbit-single-step-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r01-finite-group-orbit-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261108-r01-finite-group-orbit-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r01-finite-group-orbit-foundations/validation-report.json)