Question 23104: Subset Qualification and Subgroup Existence Are Different Converse Questions

Original question and givens

Use the complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], identity=0 and candidate_subset=[0, 1]. Row g/column h gives g*h. First certify every group axiom, then audit the entire candidate for identity, product and inverse closure; cardinality alone is not subgroup qualification. Give (group_order, candidate_subset, candidate_size, size_divides_order, candidate_is_subgroup, general_converse_claim, boundary).

• G001 — Use the complete finite Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], identity=0 and candidate_subset=[0, 1]. Row g/column h gives g*h. First certify every group axiom, then audit the entire candidate for identity, product and inverse closure; cardinality alone is not subgroup qualification. Give (group_order, candidate_subset, candidate_size, size_divides_order, candidate_is_subgroup, general_converse_claim, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-lagrange-theorem-finite-groups

Step-by-step derivation

source operation=lagrange-converse-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[group_order, candidate_subset, candidate_size, size_divides_order, candidate_is_subgroup, general_converse_claim, boundary]

ORIGINAL closed Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]];ACTUAL declared group identity=0

ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]];two-sided group identity=Yes

ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[0, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[0, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[0, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[0, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[0, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[0, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[0, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[0, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[1, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[1, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[1, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[1, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[1, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[1, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[2, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[2, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[2, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[2, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[2, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[2, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[3, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[3, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[3, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[3, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[3, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[3, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[4, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[4, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[4, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[4, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[4, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[4, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[5, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[5, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[5, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[5, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[5, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[5, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[5, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[6, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[6, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[6, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[6, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[6, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[6, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[6, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[6, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[6, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[6, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[6, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[6, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[7, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[7, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[7, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[7, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[7, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[7, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[7, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[7, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[7, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[7, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[7, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[7, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[8, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[8, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[8, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[8, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[8, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[8, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[8, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[8, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[8, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[8, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[8, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[8, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[9, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[9, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[9, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[9, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[9, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[9, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[9, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[9, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[9, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[9, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[9, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[9, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[10, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group pair(g,h)=[10, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[10, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[10, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[10, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[10, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[10, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[10, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[10, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[10, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[10, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[10, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[11, 0];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 11, 11, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, Yes], [8, 7, 7, Yes], [9, 8, 8, Yes], [10, 9, 9, Yes], [11, 10, 10, Yes]]

ORIGINAL group pair(g,h)=[11, 1];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes]]

ORIGINAL group pair(g,h)=[11, 2];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 6, 6, Yes], [6, 7, 7, Yes], [7, 8, 8, Yes], [8, 9, 9, Yes], [9, 10, 10, Yes], [10, 11, 11, Yes], [11, 0, 0, Yes]]

ORIGINAL group pair(g,h)=[11, 3];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 6, 6, Yes], [5, 7, 7, Yes], [6, 8, 8, Yes], [7, 9, 9, Yes], [8, 10, 10, Yes], [9, 11, 11, Yes], [10, 0, 0, Yes], [11, 1, 1, Yes]]

ORIGINAL group pair(g,h)=[11, 4];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 8, 8, Yes], [6, 9, 9, Yes], [7, 10, 10, Yes], [8, 11, 11, Yes], [9, 0, 0, Yes], [10, 1, 1, Yes], [11, 2, 2, Yes]]

ORIGINAL group pair(g,h)=[11, 5];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 8, 8, Yes], [5, 9, 9, Yes], [6, 10, 10, Yes], [7, 11, 11, Yes], [8, 0, 0, Yes], [9, 1, 1, Yes], [10, 2, 2, Yes], [11, 3, 3, Yes]]

ORIGINAL group pair(g,h)=[11, 6];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 8, 8, Yes], [4, 9, 9, Yes], [5, 10, 10, Yes], [6, 11, 11, Yes], [7, 0, 0, Yes], [8, 1, 1, Yes], [9, 2, 2, Yes], [10, 3, 3, Yes], [11, 4, 4, Yes]]

ORIGINAL group pair(g,h)=[11, 7];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 8, 8, Yes], [3, 9, 9, Yes], [4, 10, 10, Yes], [5, 11, 11, Yes], [6, 0, 0, Yes], [7, 1, 1, Yes], [8, 2, 2, Yes], [9, 3, 3, Yes], [10, 4, 4, Yes], [11, 5, 5, Yes]]

ORIGINAL group pair(g,h)=[11, 8];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 8, 8, Yes], [2, 9, 9, Yes], [3, 10, 10, Yes], [4, 11, 11, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 2, 2, Yes], [8, 3, 3, Yes], [9, 4, 4, Yes], [10, 5, 5, Yes], [11, 6, 6, Yes]]

ORIGINAL group pair(g,h)=[11, 9];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes], [3, 11, 11, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 3, 3, Yes], [8, 4, 4, Yes], [9, 5, 5, Yes], [10, 6, 6, Yes], [11, 7, 7, Yes]]

ORIGINAL group pair(g,h)=[11, 10];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 9, 9, Yes], [1, 10, 10, Yes], [2, 11, 11, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 4, 4, Yes], [8, 5, 5, Yes], [9, 6, 6, Yes], [10, 7, 7, Yes], [11, 8, 8, Yes]]

ORIGINAL group pair(g,h)=[11, 11];check ALL k in Fin(12)

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 10, 10, Yes], [1, 11, 11, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 5, 5, Yes], [8, 6, 6, Yes], [9, 7, 7, Yes], [10, 8, 8, Yes], [11, 9, 9, Yes]]

ORIGINAL group element g=0;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, Yes], [1, 1, 1, No], [2, 2, 2, No], [3, 3, 3, No], [4, 4, 4, No], [5, 5, 5, No], [6, 6, 6, No], [7, 7, 7, No], [8, 8, 8, No], [9, 9, 9, No], [10, 10, 10, No], [11, 11, 11, No]]

first actual two-sided inverse witness=0;exists=Yes

ORIGINAL group element g=1;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, No], [1, 2, 2, No], [2, 3, 3, No], [3, 4, 4, No], [4, 5, 5, No], [5, 6, 6, No], [6, 7, 7, No], [7, 8, 8, No], [8, 9, 9, No], [9, 10, 10, No], [10, 11, 11, No], [11, 0, 0, Yes]]

first actual two-sided inverse witness=11;exists=Yes

ORIGINAL group element g=2;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, No], [1, 3, 3, No], [2, 4, 4, No], [3, 5, 5, No], [4, 6, 6, No], [5, 7, 7, No], [6, 8, 8, No], [7, 9, 9, No], [8, 10, 10, No], [9, 11, 11, No], [10, 0, 0, Yes], [11, 1, 1, No]]

first actual two-sided inverse witness=10;exists=Yes

ORIGINAL group element g=3;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, No], [1, 4, 4, No], [2, 5, 5, No], [3, 6, 6, No], [4, 7, 7, No], [5, 8, 8, No], [6, 9, 9, No], [7, 10, 10, No], [8, 11, 11, No], [9, 0, 0, Yes], [10, 1, 1, No], [11, 2, 2, No]]

first actual two-sided inverse witness=9;exists=Yes

ORIGINAL group element g=4;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 4, 4, No], [1, 5, 5, No], [2, 6, 6, No], [3, 7, 7, No], [4, 8, 8, No], [5, 9, 9, No], [6, 10, 10, No], [7, 11, 11, No], [8, 0, 0, Yes], [9, 1, 1, No], [10, 2, 2, No], [11, 3, 3, No]]

first actual two-sided inverse witness=8;exists=Yes

ORIGINAL group element g=5;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 5, 5, No], [1, 6, 6, No], [2, 7, 7, No], [3, 8, 8, No], [4, 9, 9, No], [5, 10, 10, No], [6, 11, 11, No], [7, 0, 0, Yes], [8, 1, 1, No], [9, 2, 2, No], [10, 3, 3, No], [11, 4, 4, No]]

first actual two-sided inverse witness=7;exists=Yes

ORIGINAL group element g=6;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 6, 6, No], [1, 7, 7, No], [2, 8, 8, No], [3, 9, 9, No], [4, 10, 10, No], [5, 11, 11, No], [6, 0, 0, Yes], [7, 1, 1, No], [8, 2, 2, No], [9, 3, 3, No], [10, 4, 4, No], [11, 5, 5, No]]

first actual two-sided inverse witness=6;exists=Yes

ORIGINAL group element g=7;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 7, 7, No], [1, 8, 8, No], [2, 9, 9, No], [3, 10, 10, No], [4, 11, 11, No], [5, 0, 0, Yes], [6, 1, 1, No], [7, 2, 2, No], [8, 3, 3, No], [9, 4, 4, No], [10, 5, 5, No], [11, 6, 6, No]]

first actual two-sided inverse witness=5;exists=Yes

ORIGINAL group element g=8;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 8, 8, No], [1, 9, 9, No], [2, 10, 10, No], [3, 11, 11, No], [4, 0, 0, Yes], [5, 1, 1, No], [6, 2, 2, No], [7, 3, 3, No], [8, 4, 4, No], [9, 5, 5, No], [10, 6, 6, No], [11, 7, 7, No]]

first actual two-sided inverse witness=4;exists=Yes

ORIGINAL group element g=9;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 9, 9, No], [1, 10, 10, No], [2, 11, 11, No], [3, 0, 0, Yes], [4, 1, 1, No], [5, 2, 2, No], [6, 3, 3, No], [7, 4, 4, No], [8, 5, 5, No], [9, 6, 6, No], [10, 7, 7, No], [11, 8, 8, No]]

first actual two-sided inverse witness=3;exists=Yes

ORIGINAL group element g=10;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 10, 10, No], [1, 11, 11, No], [2, 0, 0, Yes], [3, 1, 1, No], [4, 2, 2, No], [5, 3, 3, No], [6, 4, 4, No], [7, 5, 5, No], [8, 6, 6, No], [9, 7, 7, No], [10, 8, 8, No], [11, 9, 9, No]]

first actual two-sided inverse witness=2;exists=Yes

ORIGINAL group element g=11;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 11, 11, No], [1, 0, 0, Yes], [2, 1, 1, No], [3, 2, 2, No], [4, 3, 3, No], [5, 4, 4, No], [6, 5, 5, No], [7, 6, 6, No], [8, 7, 7, No], [9, 8, 8, No], [10, 9, 9, No], [11, 10, 10, No]]

first actual two-sided inverse witness=1;exists=Yes

structurally closed COMPLETE group candidate;two-sided identity=Yes;ALL-triple associativity=Yes;ALL-element inverses=Yes

group_valid=Yes is the CONJUNCTION;False remains a reported candidate result,not a silent repair

CURRENT ORIGINAL complete Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]];actual identity=0;order=12

ALL group axioms qualified;actual inverse list=[0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1];no point carrier or group-action assumption is used

ORIGINAL candidate_subset=[0, 1];actual group identity=0

contains_identity=Yes;ALL [g,h,product,in_candidate] rows=[[0, 0, 0, Yes], [0, 1, 1, Yes], [1, 0, 1, Yes], [1, 1, 2, No]];ALL [g,inverse,in_candidate] rows=[[0, 0, Yes], [1, 11, No]];product_closed=No,inverse_closed=No,subgroup=No

ORIGINAL group order=12;candidate=[0, 1],size=2

size_divides_order=Yes;THIS candidate_is_subgroup=No;general_converse_claim fixed No;existence of SOME other subgroup of this order NOT settled

Reverse checks

• S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(12, [0, 1], 2, Yes, No, No, divisor_sized_subset_need_not_be_a_subgroup_no_existence_converse_claim)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.