Question 20145: Every Point Must Be Fixed by the Declared Identity Row

Original question and givens

Use 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] 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, 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]] with row g/column h giving g*h, declared identity=0, point carrier Fin(9)={0,…,8}, and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]] 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.

• mko-finite-group-action

Step-by-step derivation

source operation=groupaction-identity-audit;original snapshot and mathematics retained

explicit ordered source output fields=[declared_identity, point_checks, identity_action]

ORIGINAL complete group-label carrier Fin(6);declared identity label=0;Cayley table row g,column h gives g*h=[[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]]

ORIGINAL complete point carrier Fin(9);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 0, 6, 7, 8], [2, 3, 4, 5, 0, 1, 6, 7, 8], [3, 4, 5, 0, 1, 2, 6, 7, 8], [4, 5, 0, 1, 2, 3, 6, 7, 8], [5, 0, 1, 2, 3, 4, 6, 7, 8]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape

ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3, 4, 5, 6, 7, 8];complete point carrier Fin(9)

ALL ordered [point,identity_image,equality] rows=[[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, Yes]];identity-action axiom=Yes

this component audited ONLY the complete identity action row

identity-action flag alone does NOT establish group qualification or action compatibility

Reverse checks

• 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

(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, Yes], [8, 8, 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.