Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] 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 (declared_identity, point_checks, identity_action).
G001 — Use complete Cayley table=[[0, 1], [1, 0]] with row g/column h giving g*h, declared identity=0, point carrier Fin(5)={0,…,4}, and total action_table=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]] 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 (declared_identity, point_checks, identity_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.
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.
ORIGINAL complete group-label carrier Fin(2);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1], [1, 0]]
After
ORIGINAL complete point carrier Fin(5);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4], [1, 0, 2, 3, 4]];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.
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.
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.