# Question 20178: A Group Action Needs Valid Group, Identity and Compatibility Together [Return to catalogue page 101](https://ocme.evemisslab.com/en/questions/page/101/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261107-r01-finite-group-action-foundations/#q-20261014-advance-20261107-r01-finite-group-action-0003) 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. R0039-20178 · q-20261014-advance-20261107-r01-finite-group-action-0003 ## Original question and givens Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action). - G001 — Use complete Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column h giving g*h, declared identity=0, point carrier Fin(4)={0,…,3}, and total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action). ## 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-finite-group-action](https://ocme.evemisslab.com/en/mko/mko-finite-group-action/) ## Step-by-step derivation S001 · impliesBefore source operation=groupaction-qualification;original snapshot and mathematics retained After explicit ordered source output fields=[group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL complete group-label carrier Fin(4);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] After ORIGINAL complete point carrier Fin(4);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape Conditions: none The two tables have different column roles:group label h versus action point x.Complete in-range tables establish closure/totality only.A candidate group may fail its axioms;its table name does not certify them. Step identifiers and dependenciesRule: groupaction_two_table_roles_complete_candidate_not_group_assumption · Uses: G001, S001 S003 · impliesBefore ORIGINAL closed Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]];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]];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, S002 S004 · impliesBefore ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(4) 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]] 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, 1];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 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, S004 S006 · impliesBefore ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 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, S005 S007 · impliesBefore ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 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, S006 S008 · impliesBefore ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 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, S007 S009 · impliesBefore ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 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, S008 S010 · impliesBefore ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 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, S009 S011 · impliesBefore ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(4) 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]] 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)=[2, 0];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 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, S011 S013 · impliesBefore ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 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, S012 S014 · impliesBefore ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(4) 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]] 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, 3];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 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, S014 S016 · impliesBefore ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 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)=[3, 1];check ALL k in Fin(4) 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]] 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)=[3, 2];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 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, S017 S019 · impliesBefore ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(4) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 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, S018 S020 · 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]] 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, S019 S021 · 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, 0, 0, Yes]] 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, S020 S022 · 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, 0, 0, Yes], [3, 1, 1, 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, S021 S023 · impliesBefore ORIGINAL group element g=3;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, No], [1, 0, 0, Yes], [2, 1, 1, No], [3, 2, 2, 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, S022 S024 · 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, S023 S025 · impliesBefore ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3];complete point carrier Fin(4) After ALL ordered [point,identity_image,equality] rows=[[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]];identity-action axiom=Yes Conditions: none Check e*x=x at every action point,using the actual declared identity label rather than assuming0.This action axiom is distinct from the two-sided identity law in the group table. Step identifiers and dependenciesRule: groupaction_every_point_identity_action_not_group_identity · Uses: G001, S024 S026 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 0];actual group product=0;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S025 S027 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 1];actual group product=1;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S026 S028 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 2];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S027 S029 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 3];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S028 S030 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 0];actual group product=1;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S029 S031 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 1];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S030 S032 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 2];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S031 S033 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 3];actual group product=0;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S032 S034 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 0];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S033 S035 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 1];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S034 S036 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 2];actual group product=0;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S035 S037 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 3];actual group product=1;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S036 S038 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 0];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S037 S039 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 1];actual group product=0;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S038 S040 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 2];actual group product=1;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S039 S041 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 3];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S040 S042 · calculationBefore group action ordered group-pair count := mul(4, 4) After group action ordered group-pair count = 16 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore group action complete triple-domain count := mul(16, 4) After group action complete triple-domain count = 64 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · impliesBefore COMPLETE triple domain order^2*point_count=64;this derivation ACTUALLY checked all64 triples After compatibility=Yes;FIRST source-order failure[g,h,x,left,right]=null;null means NO failure anywhere Conditions: none The triple_count is the complete quantified domain size,not a measurement of an early-stopping prototype.This derivation actually checks every point for every g,h pair,so null certifies full compatibility and a first failure retains exact source data/order. Step identifiers and dependenciesRule: groupaction_full_domain_count_first_failure_not_early_stop_measurement · Uses: G001, S043 S045 · impliesBefore ORIGINAL group-row label=0;ORIGINAL point-indexed action row=[0, 1, 2, 3] After sorted image MULTISET=[0, 1, 2, 3];equals full point carrier=Yes;original row NOT reordered Conditions: none Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms. Step identifiers and dependenciesRule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S044 S046 · impliesBefore ORIGINAL group-row label=1;ORIGINAL point-indexed action row=[1, 2, 3, 0] After sorted image MULTISET=[0, 1, 2, 3];equals full point carrier=Yes;original row NOT reordered Conditions: none Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms. Step identifiers and dependenciesRule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S045 S047 · impliesBefore ORIGINAL group-row label=2;ORIGINAL point-indexed action row=[2, 3, 0, 1] After sorted image MULTISET=[0, 1, 2, 3];equals full point carrier=Yes;original row NOT reordered Conditions: none Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms. Step identifiers and dependenciesRule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S046 S048 · impliesBefore ORIGINAL group-row label=3;ORIGINAL point-indexed action row=[3, 0, 1, 2] After sorted image MULTISET=[0, 1, 2, 3];equals full point carrier=Yes;original row NOT reordered Conditions: none Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms. Step identifiers and dependenciesRule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S047 S049 · impliesBefore ACTUAL qualified group and BOTH action axioms hold;for each g actual group inverse h=[0, 3, 2, 1] After A_g composed A_h=A_(g*h)=A_e=identity and reverse composition likewise;therefore EVERY group element induces a permutation;actual row flags agree=Yes Conditions: none Use each actual group inverse and complete compatibility to make both action compositions the identity,hence every action row is a bijection.This direction derives permutations from the axioms,not validity from permutation rows alone. Step identifiers and dependenciesRule: groupaction_valid_axioms_imply_permutations_via_inverses · Uses: G001, S048 ## Reverse checks - 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 — Recompute group action complete triple-domain count with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute group action ordered group-pair count with exact rational arithmetic, not rounded decimals. - 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 (4, 4, Yes, Yes, Yes, Yes, Yes) 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: 54c4586dc1467d6e7fd9c6e80be5b2dd471a48452ac141a94717d47cdaf91cd0 - candidate record SHA-256: 1375fb0bafbffaaaed5c2764d1eb2798eb9479d66f6c746443189511418905d6 - semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034 - question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r01-finite-group-action-0003.json)