# Question 12008: Audit Every Pair of a Qualified Finite Group

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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.

R0023-12008 · q-20261013-advance-20261026-r01-finite-abelian-group-0008



## Original question and givens

Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] 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. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).


- G001 — Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 3, 5, 2, 0, 4], [2, 5, 0, 4, 3, 1], [3, 2, 4, 5, 1, 0], [4, 0, 3, 1, 5, 2], [5, 4, 1, 0, 2, 3]] 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. Give (n,identity,all noncommuting [a,b,a*b,b*a] records with a<b in label order,abelian?).





## 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](https://ocme.evemisslab.com/en/mko/mko-finite-abelian-group/)





## Step-by-step derivation

S001 · impliesBefore

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

After

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

Conditions: none

Fix the whole table and labels.Typed closure is only the first group premise;an Abelian family name does not replace the commutativity audit.

Step identifiers and dependenciesRule: complete_finite_group_typed_table · Uses: G001





S002 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S001





S003 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S002





S004 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S003





S005 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S004





S006 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S005





S007 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S006





S008 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S007





S009 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S008





S010 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S009





S011 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S010





S012 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S011





S013 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S012





S014 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S013





S015 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S014





S016 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S015





S017 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S016





S018 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S017





S019 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S018





S020 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S019





S021 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S020





S022 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S021





S023 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S022





S024 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S023





S025 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S024





S026 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S025





S027 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S026





S028 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S027





S029 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S028





S030 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S029





S031 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S030





S032 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S031





S033 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S032





S034 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S033





S035 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S034





S036 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S035





S037 · impliesBefore

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

After

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

Conditions: none

For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.

Step identifiers and dependenciesRule: associativity_every_third_label · Uses: G001, S036





S038 · impliesBefore

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

After

both equal original carrier labels=Yes

Conditions: none

Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.

Step identifiers and dependenciesRule: two_sided_identity_derive_not_zero_label · Uses: G001, S037





S039 · impliesBefore

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

After

both equal original carrier labels=No

Conditions: none

Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.

Step identifiers and dependenciesRule: two_sided_identity_derive_not_zero_label · Uses: G001, S038





S040 · impliesBefore

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

After

both equal original carrier labels=No

Conditions: none

Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.

Step identifiers and dependenciesRule: two_sided_identity_derive_not_zero_label · Uses: G001, S039





S041 · impliesBefore

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

After

both equal original carrier labels=No

Conditions: none

Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.

Step identifiers and dependenciesRule: two_sided_identity_derive_not_zero_label · Uses: G001, S040





S042 · impliesBefore

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

After

both equal original carrier labels=No

Conditions: none

Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.

Step identifiers and dependenciesRule: two_sided_identity_derive_not_zero_label · Uses: G001, S041





S043 · impliesBefore

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

After

both equal original carrier labels=No

Conditions: none

Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.

Step identifiers and dependenciesRule: two_sided_identity_derive_not_zero_label · Uses: G001, S042





S044 · impliesBefore

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

After

inverse=0

Conditions: none

Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.

Step identifiers and dependenciesRule: group_inverse_qualified_at_actual_identity · Uses: G001, S043





S045 · impliesBefore

element=1;actual identity=0;two-sided inverse candidates=[4]

After

inverse=4

Conditions: none

Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.

Step identifiers and dependenciesRule: group_inverse_qualified_at_actual_identity · Uses: G001, S044





S046 · impliesBefore

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

After

inverse=2

Conditions: none

Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.

Step identifiers and dependenciesRule: group_inverse_qualified_at_actual_identity · Uses: G001, S045





S047 · impliesBefore

element=3;actual identity=0;two-sided inverse candidates=[5]

After

inverse=5

Conditions: none

Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.

Step identifiers and dependenciesRule: group_inverse_qualified_at_actual_identity · Uses: G001, S046





S048 · impliesBefore

element=4;actual identity=0;two-sided inverse candidates=[1]

After

inverse=1

Conditions: none

Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.

Step identifiers and dependenciesRule: group_inverse_qualified_at_actual_identity · Uses: G001, S047





S049 · impliesBefore

element=5;actual identity=0;two-sided inverse candidates=[3]

After

inverse=3

Conditions: none

Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.

Step identifiers and dependenciesRule: group_inverse_qualified_at_actual_identity · Uses: G001, S048





S050 · impliesBefore

unordered distinct labels=[0, 1];a*b=1;b*a=1

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S049





S051 · impliesBefore

unordered distinct labels=[0, 2];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S050





S052 · impliesBefore

unordered distinct labels=[0, 3];a*b=3;b*a=3

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S051





S053 · impliesBefore

unordered distinct labels=[0, 4];a*b=4;b*a=4

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S052





S054 · impliesBefore

unordered distinct labels=[0, 5];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S053





S055 · impliesBefore

unordered distinct labels=[1, 2];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S054





S056 · impliesBefore

unordered distinct labels=[1, 3];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S055





S057 · impliesBefore

unordered distinct labels=[1, 4];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S056





S058 · impliesBefore

unordered distinct labels=[1, 5];a*b=4;b*a=4

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S057





S059 · impliesBefore

unordered distinct labels=[2, 3];a*b=4;b*a=4

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S058





S060 · impliesBefore

unordered distinct labels=[2, 4];a*b=3;b*a=3

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S059





S061 · impliesBefore

unordered distinct labels=[2, 5];a*b=1;b*a=1

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S060





S062 · impliesBefore

unordered distinct labels=[3, 4];a*b=1;b*a=1

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S061





S063 · impliesBefore

unordered distinct labels=[3, 5];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S062





S064 · impliesBefore

unordered distinct labels=[4, 5];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Diagonal pairs agree automatically.For each distinct pair compare both orders,retaining failures in original a<b label order.

Step identifiers and dependenciesRule: every_off_diagonal_commutativity_pair · Uses: G001, S063





S065 · impliesBefore

all noncommuting records=[]

After

specified finite group abelian=Yes

Conditions: none

All pairs must agree to make this group abelian.This classifies only the supplied finite table,not arbitrary groups.

Step identifiers and dependenciesRule: finite_abelian_all_pairs_not_general_classification · Uses: G001, S064









## Reverse checks


- S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(6, 0, [], 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: 844478419e89187c2ab14023b4f6b495e194c8179fcbfd3f9f7a4eaf4904923d
- candidate record SHA-256: 742195521289bfa6d73a3c65b06441b236afc5165c9f5dbb3598a6f62db506b9
- semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
- question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r01-finite-abelian-group-0008.json)