Question 12065: Element Order Starts from the Actual Identity

Original question and givens

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).

• G001 — Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=5. Give (element,identity,least positive order,returns to identity?).

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-abelian-group

Step-by-step derivation

carrier Fin(6);complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 4, 5, 2, 3, 0], [2, 5, 4, 1, 0, 3], [3, 2, 1, 0, 5, 4], [4, 3, 0, 5, 2, 1], [5, 0, 3, 4, 1, 2]]

all entries lie in carrier;do not assume identity0 or commutativity

a=0;b=0;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes

a=0;b=1;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[1, 4, 5, 2, 3, 0];a*(b*c) values=[1, 4, 5, 2, 3, 0];equal=Yes

a=0;b=2;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[2, 5, 4, 1, 0, 3];a*(b*c) values=[2, 5, 4, 1, 0, 3];equal=Yes

a=0;b=3;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[3, 2, 1, 0, 5, 4];a*(b*c) values=[3, 2, 1, 0, 5, 4];equal=Yes

a=0;b=4;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[4, 3, 0, 5, 2, 1];a*(b*c) values=[4, 3, 0, 5, 2, 1];equal=Yes

a=0;b=5;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[5, 0, 3, 4, 1, 2];a*(b*c) values=[5, 0, 3, 4, 1, 2];equal=Yes

a=1;b=0;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[1, 4, 5, 2, 3, 0];a*(b*c) values=[1, 4, 5, 2, 3, 0];equal=Yes

a=1;b=1;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[4, 3, 0, 5, 2, 1];a*(b*c) values=[4, 3, 0, 5, 2, 1];equal=Yes

a=1;b=2;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[5, 0, 3, 4, 1, 2];a*(b*c) values=[5, 0, 3, 4, 1, 2];equal=Yes

a=1;b=3;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[2, 5, 4, 1, 0, 3];a*(b*c) values=[2, 5, 4, 1, 0, 3];equal=Yes

a=1;b=4;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[3, 2, 1, 0, 5, 4];a*(b*c) values=[3, 2, 1, 0, 5, 4];equal=Yes

a=1;b=5;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes

a=2;b=0;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[2, 5, 4, 1, 0, 3];a*(b*c) values=[2, 5, 4, 1, 0, 3];equal=Yes

a=2;b=1;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[5, 0, 3, 4, 1, 2];a*(b*c) values=[5, 0, 3, 4, 1, 2];equal=Yes

a=2;b=2;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[4, 3, 0, 5, 2, 1];a*(b*c) values=[4, 3, 0, 5, 2, 1];equal=Yes

a=2;b=3;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[1, 4, 5, 2, 3, 0];a*(b*c) values=[1, 4, 5, 2, 3, 0];equal=Yes

a=2;b=4;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes

a=2;b=5;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[3, 2, 1, 0, 5, 4];a*(b*c) values=[3, 2, 1, 0, 5, 4];equal=Yes

a=3;b=0;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[3, 2, 1, 0, 5, 4];a*(b*c) values=[3, 2, 1, 0, 5, 4];equal=Yes

a=3;b=1;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[2, 5, 4, 1, 0, 3];a*(b*c) values=[2, 5, 4, 1, 0, 3];equal=Yes

a=3;b=2;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[1, 4, 5, 2, 3, 0];a*(b*c) values=[1, 4, 5, 2, 3, 0];equal=Yes

a=3;b=3;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes

a=3;b=4;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[5, 0, 3, 4, 1, 2];a*(b*c) values=[5, 0, 3, 4, 1, 2];equal=Yes

a=3;b=5;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[4, 3, 0, 5, 2, 1];a*(b*c) values=[4, 3, 0, 5, 2, 1];equal=Yes

a=4;b=0;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[4, 3, 0, 5, 2, 1];a*(b*c) values=[4, 3, 0, 5, 2, 1];equal=Yes

a=4;b=1;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[3, 2, 1, 0, 5, 4];a*(b*c) values=[3, 2, 1, 0, 5, 4];equal=Yes

a=4;b=2;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes

a=4;b=3;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[5, 0, 3, 4, 1, 2];a*(b*c) values=[5, 0, 3, 4, 1, 2];equal=Yes

a=4;b=4;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[2, 5, 4, 1, 0, 3];a*(b*c) values=[2, 5, 4, 1, 0, 3];equal=Yes

a=4;b=5;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[1, 4, 5, 2, 3, 0];a*(b*c) values=[1, 4, 5, 2, 3, 0];equal=Yes

a=5;b=0;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[5, 0, 3, 4, 1, 2];a*(b*c) values=[5, 0, 3, 4, 1, 2];equal=Yes

a=5;b=1;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes

a=5;b=2;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[3, 2, 1, 0, 5, 4];a*(b*c) values=[3, 2, 1, 0, 5, 4];equal=Yes

a=5;b=3;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[4, 3, 0, 5, 2, 1];a*(b*c) values=[4, 3, 0, 5, 2, 1];equal=Yes

a=5;b=4;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[1, 4, 5, 2, 3, 0];a*(b*c) values=[1, 4, 5, 2, 3, 0];equal=Yes

a=5;b=5;all c=[0, 1, 2, 3, 4, 5]

(a*b)*c values=[2, 5, 4, 1, 0, 3];a*(b*c) values=[2, 5, 4, 1, 0, 3];equal=Yes

candidate identity=0;left products=[0, 1, 2, 3, 4, 5];right products=[0, 1, 2, 3, 4, 5]

both equal original carrier labels=Yes

candidate identity=1;left products=[1, 4, 5, 2, 3, 0];right products=[1, 4, 5, 2, 3, 0]

both equal original carrier labels=No

candidate identity=2;left products=[2, 5, 4, 1, 0, 3];right products=[2, 5, 4, 1, 0, 3]

both equal original carrier labels=No

candidate identity=3;left products=[3, 2, 1, 0, 5, 4];right products=[3, 2, 1, 0, 5, 4]

both equal original carrier labels=No

candidate identity=4;left products=[4, 3, 0, 5, 2, 1];right products=[4, 3, 0, 5, 2, 1]

both equal original carrier labels=No

candidate identity=5;left products=[5, 0, 3, 4, 1, 2];right products=[5, 0, 3, 4, 1, 2]

both equal original carrier labels=No

element=0;actual identity=0;two-sided inverse candidates=[0]

inverse=0

element=1;actual identity=0;two-sided inverse candidates=[5]

inverse=5

element=2;actual identity=0;two-sided inverse candidates=[4]

inverse=4

element=3;actual identity=0;two-sided inverse candidates=[3]

inverse=3

element=4;actual identity=0;two-sided inverse candidates=[2]

inverse=2

element=5;actual identity=0;two-sided inverse candidates=[1]

inverse=1

element=5;actual identity=0

zero repetitions give=0

repetition=1;previous=0;element=5;table row=[0, 1, 2, 3, 4, 5]

new value=table[0][5]=5

repetition=2;previous=5;element=5;table row=[5, 0, 3, 4, 1, 2]

new value=table[5][5]=2

repetition=3;previous=2;element=5;table row=[2, 5, 4, 1, 0, 3]

new value=table[2][5]=3

repetition=4;previous=3;element=5;table row=[3, 2, 1, 0, 5, 4]

new value=table[3][5]=4

repetition=5;previous=4;element=5;table row=[4, 3, 0, 5, 2, 1]

new value=table[4][5]=1

repetition=6;previous=1;element=5;table row=[1, 4, 5, 2, 3, 0]

new value=table[1][5]=0

successive distinct powers=[0, 5, 2, 3, 4, 1];first repeat=0;actual identity=0

least positive return order=6;generated cyclic subgroup size=6

Reverse checks

• 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

(5, 0, 6, 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.