Complete Group Orbits, Equivalence Witnesses and Partition Certificates

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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