# Question 23137: The Index Times Subgroup Order Equals Group Order

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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.

R0045-23137 · q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0062



## Original question and givens

Use the complete finite Cayley table=[[6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4]], identity=2 and subset=[0, 2, 4, 6]. 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 (group_order, subgroup_order, index, product, remainder, equal).


- G001 — Use the complete finite Cayley table=[[6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4]], identity=2 and subset=[0, 2, 4, 6]. 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 (group_order, subgroup_order, index, product, remainder, equal).





## 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](https://ocme.evemisslab.com/en/mko/mko-lagrange-theorem-finite-groups/)





## Step-by-step derivation

S001 · impliesBefore

source operation=lagrange-divisibility-identity;original snapshot and mathematics retained

After

explicit ordered source output fields=[group_order, subgroup_order, index, product, remainder, equal]

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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL closed Cayley table=[[6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4]];ACTUAL declared group identity=2

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_group_identity_both_sides_not_action_identity · Uses: G001, S001





S003 · impliesBefore

ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S002





S004 · impliesBefore

ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S003





S005 · impliesBefore

ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S004





S006 · impliesBefore

ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S005





S007 · impliesBefore

ORIGINAL group pair(g,h)=[0, 4];check ALL k in Fin(8)

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S006





S008 · impliesBefore

ORIGINAL group pair(g,h)=[0, 5];check ALL k in Fin(8)

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, 6, 6, Yes], [6, 7, 7, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S007





S009 · impliesBefore

ORIGINAL group pair(g,h)=[0, 6];check ALL k in Fin(8)

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, 6, 6, Yes], [5, 7, 7, Yes], [6, 0, 0, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S008





S010 · impliesBefore

ORIGINAL group pair(g,h)=[0, 7];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S009





S011 · impliesBefore

ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S010





S012 · impliesBefore

ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S011





S013 · impliesBefore

ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S012





S014 · impliesBefore

ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(8)

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S013





S015 · impliesBefore

ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(8)

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, 6, 6, Yes], [6, 7, 7, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S014





S016 · impliesBefore

ORIGINAL group pair(g,h)=[1, 5];check ALL k in Fin(8)

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, 6, 6, Yes], [5, 7, 7, Yes], [6, 0, 0, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S015





S017 · impliesBefore

ORIGINAL group pair(g,h)=[1, 6];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S016





S018 · impliesBefore

ORIGINAL group pair(g,h)=[1, 7];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S017





S019 · impliesBefore

ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S018





S020 · impliesBefore

ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S019





S021 · impliesBefore

ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(8)

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S020





S022 · impliesBefore

ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(8)

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, 6, 6, Yes], [6, 7, 7, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S021





S023 · impliesBefore

ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(8)

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, 6, 6, Yes], [5, 7, 7, Yes], [6, 0, 0, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S022





S024 · impliesBefore

ORIGINAL group pair(g,h)=[2, 5];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S023





S025 · impliesBefore

ORIGINAL group pair(g,h)=[2, 6];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S024





S026 · impliesBefore

ORIGINAL group pair(g,h)=[2, 7];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S025





S027 · impliesBefore

ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S026





S028 · impliesBefore

ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(8)

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S027





S029 · impliesBefore

ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(8)

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, 6, 6, Yes], [6, 7, 7, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S028





S030 · impliesBefore

ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(8)

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, 6, 6, Yes], [5, 7, 7, Yes], [6, 0, 0, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S029





S031 · impliesBefore

ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S030





S032 · impliesBefore

ORIGINAL group pair(g,h)=[3, 5];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S031





S033 · impliesBefore

ORIGINAL group pair(g,h)=[3, 6];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S032





S034 · impliesBefore

ORIGINAL group pair(g,h)=[3, 7];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S033





S035 · impliesBefore

ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(8)

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S034





S036 · impliesBefore

ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(8)

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, 6, 6, Yes], [6, 7, 7, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S035





S037 · impliesBefore

ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(8)

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, 6, 6, Yes], [5, 7, 7, Yes], [6, 0, 0, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S036





S038 · impliesBefore

ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S037





S039 · impliesBefore

ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S038





S040 · impliesBefore

ORIGINAL group pair(g,h)=[4, 5];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S039





S041 · impliesBefore

ORIGINAL group pair(g,h)=[4, 6];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S040





S042 · impliesBefore

ORIGINAL group pair(g,h)=[4, 7];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S041





S043 · impliesBefore

ORIGINAL group pair(g,h)=[5, 0];check ALL k in Fin(8)

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, 6, 6, Yes], [6, 7, 7, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S042





S044 · impliesBefore

ORIGINAL group pair(g,h)=[5, 1];check ALL k in Fin(8)

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, 6, 6, Yes], [5, 7, 7, Yes], [6, 0, 0, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S043





S045 · impliesBefore

ORIGINAL group pair(g,h)=[5, 2];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S044





S046 · impliesBefore

ORIGINAL group pair(g,h)=[5, 3];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S045





S047 · impliesBefore

ORIGINAL group pair(g,h)=[5, 4];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S046





S048 · impliesBefore

ORIGINAL group pair(g,h)=[5, 5];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S047





S049 · impliesBefore

ORIGINAL group pair(g,h)=[5, 6];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S048





S050 · impliesBefore

ORIGINAL group pair(g,h)=[5, 7];check ALL k in Fin(8)

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S049





S051 · impliesBefore

ORIGINAL group pair(g,h)=[6, 0];check ALL k in Fin(8)

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, 6, 6, Yes], [5, 7, 7, Yes], [6, 0, 0, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S050





S052 · impliesBefore

ORIGINAL group pair(g,h)=[6, 1];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S051





S053 · impliesBefore

ORIGINAL group pair(g,h)=[6, 2];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S052





S054 · impliesBefore

ORIGINAL group pair(g,h)=[6, 3];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S053





S055 · impliesBefore

ORIGINAL group pair(g,h)=[6, 4];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S054





S056 · impliesBefore

ORIGINAL group pair(g,h)=[6, 5];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S055





S057 · impliesBefore

ORIGINAL group pair(g,h)=[6, 6];check ALL k in Fin(8)

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S056





S058 · impliesBefore

ORIGINAL group pair(g,h)=[6, 7];check ALL k in Fin(8)

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, 6, 6, Yes], [6, 7, 7, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S057





S059 · impliesBefore

ORIGINAL group pair(g,h)=[7, 0];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes], [3, 6, 6, Yes], [4, 7, 7, Yes], [5, 0, 0, Yes], [6, 1, 1, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S058





S060 · impliesBefore

ORIGINAL group pair(g,h)=[7, 1];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 5, 5, Yes], [2, 6, 6, Yes], [3, 7, 7, Yes], [4, 0, 0, Yes], [5, 1, 1, Yes], [6, 2, 2, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S059





S061 · impliesBefore

ORIGINAL group pair(g,h)=[7, 2];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 5, 5, Yes], [1, 6, 6, Yes], [2, 7, 7, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 2, 2, Yes], [6, 3, 3, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S060





S062 · impliesBefore

ORIGINAL group pair(g,h)=[7, 3];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 6, 6, Yes], [1, 7, 7, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 3, 3, Yes], [6, 4, 4, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S061





S063 · impliesBefore

ORIGINAL group pair(g,h)=[7, 4];check ALL k in Fin(8)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 7, 7, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 4, 4, Yes], [6, 5, 5, Yes], [7, 6, 6, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S062





S064 · impliesBefore

ORIGINAL group pair(g,h)=[7, 5];check ALL k in Fin(8)

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], [6, 6, 6, Yes], [7, 7, 7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S063





S065 · impliesBefore

ORIGINAL group pair(g,h)=[7, 6];check ALL k in Fin(8)

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, 6, 6, Yes], [6, 7, 7, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S064





S066 · impliesBefore

ORIGINAL group pair(g,h)=[7, 7];check ALL k in Fin(8)

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, 6, 6, Yes], [5, 7, 7, Yes], [6, 0, 0, Yes], [7, 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 dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S065





S067 · impliesBefore

ORIGINAL group element g=0;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 6, 6, No], [1, 7, 7, No], [2, 0, 0, No], [3, 1, 1, No], [4, 2, 2, Yes], [5, 3, 3, No], [6, 4, 4, No], [7, 5, 5, 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 dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S066





S068 · impliesBefore

ORIGINAL group element g=1;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 7, 7, No], [1, 0, 0, No], [2, 1, 1, No], [3, 2, 2, Yes], [4, 3, 3, No], [5, 4, 4, No], [6, 5, 5, No], [7, 6, 6, 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 dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S067





S069 · impliesBefore

ORIGINAL group element g=2;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, No], [1, 1, 1, No], [2, 2, 2, Yes], [3, 3, 3, No], [4, 4, 4, No], [5, 5, 5, No], [6, 6, 6, No], [7, 7, 7, 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 dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S068





S070 · impliesBefore

ORIGINAL group element g=3;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, No], [1, 2, 2, Yes], [2, 3, 3, No], [3, 4, 4, No], [4, 5, 5, No], [5, 6, 6, No], [6, 7, 7, No], [7, 0, 0, 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 dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S069





S071 · impliesBefore

ORIGINAL group element g=4;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, Yes], [1, 3, 3, No], [2, 4, 4, No], [3, 5, 5, No], [4, 6, 6, No], [5, 7, 7, No], [6, 0, 0, No], [7, 1, 1, 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 dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S070





S072 · impliesBefore

ORIGINAL group element g=5;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, No], [1, 4, 4, No], [2, 5, 5, No], [3, 6, 6, No], [4, 7, 7, No], [5, 0, 0, No], [6, 1, 1, No], [7, 2, 2, Yes]]

After

first actual two-sided inverse witness=7;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 dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S071





S073 · impliesBefore

ORIGINAL group element g=6;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 4, 4, No], [1, 5, 5, No], [2, 6, 6, No], [3, 7, 7, No], [4, 0, 0, No], [5, 1, 1, No], [6, 2, 2, Yes], [7, 3, 3, No]]

After

first actual two-sided inverse witness=6;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 dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S072





S074 · impliesBefore

ORIGINAL group element g=7;DECLARED identity=2;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 5, 5, No], [1, 6, 6, No], [2, 7, 7, No], [3, 0, 0, No], [4, 1, 1, No], [5, 2, 2, Yes], [6, 3, 3, No], [7, 4, 4, No]]

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 dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S073





S075 · impliesBefore

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 dependenciesRule: groupaction_group_axiom_conjunction_candidate_may_fail · Uses: G001, S074





S076 · impliesBefore

CURRENT ORIGINAL complete Cayley table=[[6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4]];actual identity=2;order=8

After

ALL group axioms qualified;actual inverse list=[4, 3, 2, 1, 0, 7, 6, 5];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 dependenciesRule: group_CURRENT_complete_certified_Cayley_NOT_action_wrapper · Uses: G001, S075





S077 · impliesBefore

ORIGINAL subset=[0, 2, 4, 6];actual group identity=2

After

contains_identity=Yes;ALL [g,h,product,in_candidate] rows=[[0, 0, 6, Yes], [0, 2, 0, Yes], [0, 4, 2, Yes], [0, 6, 4, Yes], [2, 0, 0, Yes], [2, 2, 2, Yes], [2, 4, 4, Yes], [2, 6, 6, Yes], [4, 0, 2, Yes], [4, 2, 4, Yes], [4, 4, 6, Yes], [4, 6, 0, Yes], [6, 0, 4, Yes], [6, 2, 6, Yes], [6, 4, 0, Yes], [6, 6, 2, Yes]];ALL [g,inverse,in_candidate] rows=[[0, 4, Yes], [2, 2, Yes], [4, 0, Yes], [6, 6, 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 dependenciesRule: lagrange_candidate_ALL_subgroup_obligations_not_size · Uses: G001, S076





S078 · impliesBefore

ORIGINAL group representative g=0;SAME subgroup H=[0, 2, 4, 6];inverse of g=4

After

ALL [h,g*h] rows=[[0, 6], [2, 0], [4, 2], [6, 4]];LEFT coset=[0, 2, 4, 6];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 2], [2, 4], [4, 6], [6, 0]];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 dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S077





S079 · impliesBefore

ORIGINAL group representative g=1;SAME subgroup H=[0, 2, 4, 6];inverse of g=3

After

ALL [h,g*h] rows=[[0, 7], [2, 1], [4, 3], [6, 5]];LEFT coset=[1, 3, 5, 7];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 2], [3, 4], [5, 6], [7, 0]];least representative for this distinct set=1

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 dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S078





S080 · impliesBefore

ORIGINAL group representative g=2;SAME subgroup H=[0, 2, 4, 6];inverse of g=2

After

ALL [h,g*h] rows=[[0, 0], [2, 2], [4, 4], [6, 6]];LEFT coset=[0, 2, 4, 6];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 0], [2, 2], [4, 4], [6, 6]];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 dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S079





S081 · impliesBefore

ORIGINAL group representative g=3;SAME subgroup H=[0, 2, 4, 6];inverse of g=1

After

ALL [h,g*h] rows=[[0, 1], [2, 3], [4, 5], [6, 7]];LEFT coset=[1, 3, 5, 7];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 0], [3, 2], [5, 4], [7, 6]];least representative for this distinct set=1

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 dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S080





S082 · impliesBefore

ORIGINAL group representative g=4;SAME subgroup H=[0, 2, 4, 6];inverse of g=0

After

ALL [h,g*h] rows=[[0, 2], [2, 4], [4, 6], [6, 0]];LEFT coset=[0, 2, 4, 6];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 6], [2, 0], [4, 2], [6, 4]];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 dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S081





S083 · impliesBefore

ORIGINAL group representative g=5;SAME subgroup H=[0, 2, 4, 6];inverse of g=7

After

ALL [h,g*h] rows=[[0, 3], [2, 5], [4, 7], [6, 1]];LEFT coset=[1, 3, 5, 7];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 6], [3, 0], [5, 2], [7, 4]];least representative for this distinct set=1

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 dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S082





S084 · impliesBefore

ORIGINAL group representative g=6;SAME subgroup H=[0, 2, 4, 6];inverse of g=6

After

ALL [h,g*h] rows=[[0, 4], [2, 6], [4, 0], [6, 2]];LEFT coset=[0, 2, 4, 6];ALL [g*h,g_inverse*(g*h)] reverse rows=[[0, 4], [2, 6], [4, 0], [6, 2]];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 dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S083





S085 · impliesBefore

ORIGINAL group representative g=7;SAME subgroup H=[0, 2, 4, 6];inverse of g=5

After

ALL [h,g*h] rows=[[0, 5], [2, 7], [4, 1], [6, 3]];LEFT coset=[1, 3, 5, 7];ALL [g*h,g_inverse*(g*h)] reverse rows=[[1, 4], [3, 6], [5, 0], [7, 2]];least representative for this distinct set=1

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 dependenciesRule: lagrange_EVERY_actual_LEFT_translation_two_way_bijection · Uses: G001, S084





S086 · impliesBefore

ALL distinct LEFT cosets sorted by SOURCE English element-list order=[[0, [0, 2, 4, 6]], [1, [1, 3, 5, 7]]]

After

every size=4;flattened elements=[0, 1, 2, 3, 4, 5, 6, 7];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 dependenciesRule: lagrange_LEFT_cosets_equal_or_disjoint_cover_with_source_order · Uses: G001, S085





S087 · calculationBefore

Lagrange subgroup order times COMPLETE coset index := mul(4, 2)

After

Lagrange subgroup order times COMPLETE coset index = 8

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086









## Reverse checks


- S087 · premise_and_result_audit — Recompute Lagrange subgroup order times COMPLETE coset index with exact rational arithmetic, not rounded decimals.
- S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(8, 4, 2, 8, 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: 401370ab3d401016a6eceab7b527920846775f63a32d3a3d2cb0e65f1a6132f8
- candidate record SHA-256: d5806c09ea5c70fc1ac7e39d88fd11a60fb76b56f0a676ab6f405210c97dc838
- semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
- question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r01-lagrange-theorem-finite-groups-0062.json)