Question 23166: Complete Left Cosets Are Equal or Disjoint and Cover the Group

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0045-23166 · q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0041

Original question and givens

Use the complete finite Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]], identity=3 and subset=[1, 3, 5]. Row g/column h gives g*h. First certify every group axiom, then audit the entire candidate for identity, product and inverse closure; cardinality alone is not subgroup qualification. Give (representative_coset_rows, equal_sizes, flattened_elements, partition_certificate, certificate).

G001 — Use the complete finite Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]], identity=3 and subset=[1, 3, 5]. Row g/column h gives g*h. First certify every group axiom, then audit the entire candidate for identity, product and inverse closure; cardinality alone is not subgroup qualification. Give (representative_coset_rows, equal_sizes, flattened_elements, partition_certificate, certificate).

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-lagrange-theorem-finite-groups

Step-by-step derivation

S001 · implies

Before

source operation=lagrange-left-coset-partition;original snapshot and mathematics retained

After

explicit ordered source output fields=[representative_coset_rows, equal_sizes, flattened_elements, partition_certificate, certificate]

Conditions: none

Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL closed Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]];ACTUAL declared group identity=3

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], [4, 4, 4, Yes], [5, 5, 5, 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 dependencies

Rule: groupaction_group_identity_both_sides_not_action_identity · Uses: G001, S001

S003 · implies

Before

ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(6)

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], [4, 4, 4, Yes], [5, 5, 5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S002

S004 · implies

Before

ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S003

S005 · implies

Before

ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 0, 0, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S004

S006 · implies

Before

ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S005

S007 · implies

Before

ORIGINAL group pair(g,h)=[0, 4];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S006

S008 · implies

Before

ORIGINAL group pair(g,h)=[0, 5];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S007

S009 · implies

Before

ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S008

S010 · implies

Before

ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 0, 0, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S009

S011 · implies

Before

ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S010

S012 · implies

Before

ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S011

S013 · implies

Before

ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S012

S014 · implies

Before

ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(6)

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], [4, 4, 4, Yes], [5, 5, 5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S013

S015 · implies

Before

ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 0, 0, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S014

S016 · implies

Before

ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S015

S017 · implies

Before

ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S016

S018 · implies

Before

ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S017

S019 · implies

Before

ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(6)

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], [4, 4, 4, Yes], [5, 5, 5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S018

S020 · implies

Before

ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S019

S021 · implies

Before

ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S020

S022 · implies

Before

ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S021

S023 · implies

Before

ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S022

S024 · implies

Before

ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(6)

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], [4, 4, 4, Yes], [5, 5, 5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S023

S025 · implies

Before

ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S024

S026 · implies

Before

ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 0, 0, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S025

S027 · implies

Before

ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S026

S028 · implies

Before

ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S027

S029 · implies

Before

ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(6)

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], [4, 4, 4, Yes], [5, 5, 5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S028

S030 · implies

Before

ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S029

S031 · implies

Before

ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 0, 0, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S030

S032 · implies

Before

ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S031

S033 · implies

Before

ORIGINAL group pair(g,h)=[5, 0];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S032

S034 · implies

Before

ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(6)

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], [4, 4, 4, Yes], [5, 5, 5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S033

S035 · implies

Before

ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 5, 5, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S034

S036 · implies

Before

ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 5, 5, Yes], [4, 0, 0, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S035

S037 · implies

Before

ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S036

S038 · implies

Before

ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(6)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S037

S039 · implies

Before

ORIGINAL group element g=0;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, Yes], [1, 4, 4, No], [2, 5, 5, No], [3, 0, 0, No], [4, 1, 1, No], [5, 2, 2, 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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S038

S040 · implies

Before

ORIGINAL group element g=1;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 4, 4, No], [1, 5, 5, No], [2, 0, 0, No], [3, 1, 1, No], [4, 2, 2, No], [5, 3, 3, Yes]]

After

first actual two-sided inverse witness=5;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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S039

S041 · implies

Before

ORIGINAL group element g=2;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 5, 5, No], [1, 0, 0, No], [2, 1, 1, No], [3, 2, 2, No], [4, 3, 3, Yes], [5, 4, 4, No]]

After

first actual two-sided inverse witness=4;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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S040

S042 · implies

Before

ORIGINAL group element g=3;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, No], [1, 1, 1, No], [2, 2, 2, No], [3, 3, 3, Yes], [4, 4, 4, No], [5, 5, 5, No]]

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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S041

S043 · implies

Before

ORIGINAL group element g=4;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, No], [1, 2, 2, No], [2, 3, 3, Yes], [3, 4, 4, No], [4, 5, 5, No], [5, 0, 0, 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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S042

S044 · implies

Before

ORIGINAL group element g=5;DECLARED identity=3;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, No], [1, 3, 3, Yes], [2, 4, 4, No], [3, 5, 5, No], [4, 0, 0, No], [5, 1, 1, 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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S043

S045 · implies

Before

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 dependencies

Rule: groupaction_group_axiom_conjunction_candidate_may_fail · Uses: G001, S044

S046 · implies

Before

CURRENT ORIGINAL complete Cayley table=[[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1]];actual identity=3;order=6

After

ALL group axioms qualified;actual inverse list=[0, 5, 4, 3, 2, 1];no point carrier or group-action assumption is used

Conditions: none

Certify closure,identity,associativity and two-sided inverses in the group table itself.Do not import an old8-element action-table limit or assume cyclicity/commutativity.The frozen pure helper receives only this question's table and trace.

Step identifiers and dependencies

Rule: group_CURRENT_complete_certified_Cayley_NOT_action_wrapper · Uses: G001, S045

S047 · implies

Before

ORIGINAL subset=[1, 3, 5];actual group identity=3

After

contains_identity=Yes;ALL [g,h,product,in_candidate] rows=[[1, 1, 5, Yes], [1, 3, 1, Yes], [1, 5, 3, Yes], [3, 1, 1, Yes], [3, 3, 3, Yes], [3, 5, 5, Yes], [5, 1, 3, Yes], [5, 3, 5, Yes], [5, 5, 1, Yes]];ALL [g,inverse,in_candidate] rows=[[1, 5, Yes], [3, 3, Yes], [5, 1, Yes]];product_closed=Yes,inverse_closed=Yes,subgroup=Yes

Conditions: none

A complete candidate may fail.Identity,product and inverse closure must all hold;divisor-sized cardinality replaces none of them.

Step identifiers and dependencies

Rule: lagrange_candidate_ALL_subgroup_obligations_not_size · Uses: G001, S046

S048 · implies

Before

ORIGINAL group representative g=0;SAME subgroup H=[1, 3, 5];inverse of g=0

After

ALL [h,g*h] rows=[[1, 4], [3, 0], [5, 2]];LEFT coset=[0, 2, 4];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 3], [2, 5], [4, 1]];least representative for this distinct set=0

Conditions: none

Left translation h->g*h and u->g_inverse*u are inverse maps,so every left coset has H's size.Keep the first original g as its representative;do not substitute a right coset or another equally sized set.

Step identifiers and dependencies

Rule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S047

S049 · implies

Before

ORIGINAL group representative g=1;SAME subgroup H=[1, 3, 5];inverse of g=5

After

ALL [h,g*h] rows=[[1, 5], [3, 1], [5, 3]];LEFT coset=[1, 3, 5];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 3], [3, 5], [5, 1]];least representative for this distinct set=1

Conditions: none

Step identifiers and dependencies

Rule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S048

S050 · implies

Before

ORIGINAL group representative g=2;SAME subgroup H=[1, 3, 5];inverse of g=4

After

ALL [h,g*h] rows=[[1, 0], [3, 2], [5, 4]];LEFT coset=[0, 2, 4];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 1], [2, 3], [4, 5]];least representative for this distinct set=0

Conditions: none

Step identifiers and dependencies

Rule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S049

S051 · implies

Before

ORIGINAL group representative g=3;SAME subgroup H=[1, 3, 5];inverse of g=3

After

ALL [h,g*h] rows=[[1, 1], [3, 3], [5, 5]];LEFT coset=[1, 3, 5];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 1], [3, 3], [5, 5]];least representative for this distinct set=1

Conditions: none

Step identifiers and dependencies

Rule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S050

S052 · implies

Before

ORIGINAL group representative g=4;SAME subgroup H=[1, 3, 5];inverse of g=2

After

ALL [h,g*h] rows=[[1, 2], [3, 4], [5, 0]];LEFT coset=[0, 2, 4];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 5], [2, 1], [4, 3]];least representative for this distinct set=0

Conditions: none

Step identifiers and dependencies

Rule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S051

S053 · implies

Before

ORIGINAL group representative g=5;SAME subgroup H=[1, 3, 5];inverse of g=1

After

ALL [h,g*h] rows=[[1, 3], [3, 5], [5, 1]];LEFT coset=[1, 3, 5];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 5], [3, 1], [5, 3]];least representative for this distinct set=1

Conditions: none

Step identifiers and dependencies

Rule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S052

S054 · implies

Before

ALL distinct LEFT cosets sorted by SOURCE English element-list order=[[0, [0, 2, 4]], [1, [1, 3, 5]]]

After

every size=3;flattened elements=[0, 1, 2, 3, 4, 5];partition covers ORIGINAL group exactly once=Yes

Conditions: none

If gh=kh1,then k_inverse*g=h1*h_inverse lies in H and the two cosets contain each other;otherwise they are disjoint.The identity puts each g in gH,so they cover G.Sort only by the coset element list,not the representative field.

Step identifiers and dependencies

Rule: lagrange_LEFT_cosets_equal_or_disjoint_cover_with_source_order · Uses: G001, S053

Reverse checks

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

([[0, [0, 2, 4]], [1, [1, 3, 5]]], Yes, [0, 1, 2, 3, 4, 5], Yes, left_cosets_equal_or_disjoint_and_cover_G)

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: cee17421d7f29ebad68cca195ca2779826483c30768cb6eb5daa91244c8eebbb
candidate record SHA-256: d21e034b7dd9a1908b97471e1d0de418035c38d01aefa7e39d944c17f5559e2b
semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON