# Question 23101: Subset Qualification and Subgroup Existence Are Different Converse Questions [Return to catalogue page 116](https://ocme.evemisslab.com/en/questions/page/116/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-foundations/#q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0076) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0045-23101 · q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0076 ## Original question and givens Use the complete finite Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], 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], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0 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](https://ocme.evemisslab.com/en/mko/mko-lagrange-theorem-finite-groups/) ## Step-by-step derivation S001 · impliesBefore source operation=lagrange-converse-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[group_order, candidate_subset, candidate_size, size_divides_order, candidate_is_subgroup, general_converse_claim, boundary] Conditions: none Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL closed Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]];ACTUAL declared group identity=0 After 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]];two-sided group identity=Yes Conditions: none Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points. Step identifiers and dependenciesRule: groupaction_group_identity_both_sides_not_action_identity · Uses: G001, S001 S003 · impliesBefore ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(6) After 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]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S002 S004 · impliesBefore ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(6) After 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, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S003 S005 · impliesBefore ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(6) After 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, 0, 0, Yes], [5, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S004 S006 · impliesBefore ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S005 S007 · impliesBefore ORIGINAL group pair(g,h)=[0, 4];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S006 S008 · impliesBefore ORIGINAL group pair(g,h)=[0, 5];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S007 S009 · impliesBefore ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(6) After 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, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S008 S010 · impliesBefore ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(6) After 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, 0, 0, Yes], [5, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S009 S011 · impliesBefore ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S010 S012 · impliesBefore ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S011 S013 · impliesBefore ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S012 S014 · impliesBefore ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(6) After 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]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S013 S015 · impliesBefore ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(6) After 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, 0, 0, Yes], [5, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S014 S016 · impliesBefore ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S015 S017 · impliesBefore ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S016 S018 · impliesBefore ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S017 S019 · impliesBefore ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(6) After 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]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S018 S020 · impliesBefore ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(6) After 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, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S019 S021 · impliesBefore ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S020 S022 · impliesBefore ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S021 S023 · impliesBefore ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S022 S024 · impliesBefore ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(6) After 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]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S023 S025 · impliesBefore ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(6) After 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, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S024 S026 · impliesBefore ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(6) After 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, 0, 0, Yes], [5, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S025 S027 · impliesBefore ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S026 S028 · impliesBefore ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S027 S029 · impliesBefore ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(6) After 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]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S028 S030 · impliesBefore ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(6) After 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, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S029 S031 · impliesBefore ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(6) After 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, 0, 0, Yes], [5, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S030 S032 · impliesBefore ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S031 S033 · impliesBefore ORIGINAL group pair(g,h)=[5, 0];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S032 S034 · impliesBefore ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(6) After 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]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S033 S035 · impliesBefore ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(6) After 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, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S034 S036 · impliesBefore ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(6) After 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, 0, 0, Yes], [5, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S035 S037 · impliesBefore ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S036 S038 · impliesBefore ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(6) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S037 S039 · impliesBefore 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]] After first actual two-sided inverse witness=0;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S038 S040 · impliesBefore 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, 0, 0, Yes]] After first actual two-sided inverse witness=5;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S039 S041 · impliesBefore 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, 0, 0, Yes], [5, 1, 1, No]] After first actual two-sided inverse witness=4;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S040 S042 · impliesBefore 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, 0, 0, Yes], [4, 1, 1, No], [5, 2, 2, No]] After first actual two-sided inverse witness=3;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S041 S043 · impliesBefore 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, 0, 0, Yes], [3, 1, 1, No], [4, 2, 2, No], [5, 3, 3, No]] After first actual two-sided inverse witness=2;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S042 S044 · impliesBefore ORIGINAL group element g=5;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 5, 5, No], [1, 0, 0, Yes], [2, 1, 1, No], [3, 2, 2, No], [4, 3, 3, No], [5, 4, 4, No]] After first actual two-sided inverse witness=1;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S043 S045 · impliesBefore structurally closed COMPLETE group candidate;two-sided identity=Yes;ALL-triple associativity=Yes;ALL-element inverses=Yes After group_valid=Yes is the CONJUNCTION;False remains a reported candidate result,not a silent repair Conditions: none Group qualification is the conjunction of these complete obligations.A permutation-looking action table cannot repair a bad group table;do not replace the identity or modify the table. Step identifiers and dependenciesRule: groupaction_group_axiom_conjunction_candidate_may_fail · Uses: G001, S044 S046 · impliesBefore CURRENT ORIGINAL complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]];actual identity=0;order=6 After ALL group axioms qualified;actual inverse list=[0, 5, 4, 3, 2, 1];no point carrier or group-action assumption is used Conditions: none Certify closure,identity,associativity and two-sided inverses in the group table itself.Do not import an old8-element action-table limit or assume cyclicity/commutativity.The frozen pure helper receives only this question's table and trace. Step identifiers and dependenciesRule: group_CURRENT_complete_certified_Cayley_NOT_action_wrapper · Uses: G001, S045 S047 · impliesBefore ORIGINAL candidate_subset=[0, 1];actual group identity=0 After 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, 5, No]];product_closed=No,inverse_closed=No,subgroup=No Conditions: none A complete candidate may fail.Identity,product and inverse closure must all hold;divisor-sized cardinality replaces none of them. Step identifiers and dependenciesRule: lagrange_candidate_ALL_subgroup_obligations_not_size · Uses: G001, S046 S048 · impliesBefore ORIGINAL group order=6;candidate=[0, 1],size=2 After size_divides_order=Yes;THIS candidate_is_subgroup=No;general_converse_claim fixed No;existence of SOME other subgroup of this order NOT settled Conditions: none The No negates neither a true candidate flag nor the existence of another subgroup of the same order.It only withholds a general converse from cardinality alone. Step identifiers and dependenciesRule: lagrange_candidate_vs_existence_general_converse_No · Uses: G001, S047 ## Reverse checks - 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 (6, [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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 6f3b671358be432f250db8ed0aa0ea4d401a326d159e7d5f1e479f6b76d56f35 - candidate record SHA-256: fc25c49b7cbf1d67bce4b2b2a86f7f8a86b2de734e70b54d603a12f687d018c1 - semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438 - question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0076.json)