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Question 24544: Left and Right Cosets Must Agree for Every g

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0048-24544 · q-20261015-advance-20261113-r01-finite-normal-subgroup-0069

Original question and givens

Use complete finite Cayley table=[[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]], identity=3 and candidate subset H=[3]. Row g/column h gives g*h. Validate all group axioms, then identity/product/inverse closure of the whole subset before treating it as a subgroup. Normality has a separate all-g obligation. Give (subgroup, coset_rows, all_left_right_equal, certificate).

  • G001 — Use complete finite Cayley table=[[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]], identity=3 and candidate subset H=[3]. Row g/column h gives g*h. Validate all group axioms, then identity/product/inverse closure of the whole subset before treating it as a subgroup. Normality has a separate all-g obligation. Give (subgroup, coset_rows, all_left_right_equal, 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.

Step-by-step derivation

S001 · implies

Before

source roles = [normal-left-right-coset-audit]

After

derived = [[subgroup, coset_rows, all_left_right_equal, certificate]]

Conditions: none

List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [Normality ORIGINAL group, 32, [[7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5]], 3]

After

derived = [10, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9]], Yes]

Conditions: none

Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables.

Step identifiers and dependencies

Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001

S003 · implies

Before

source roles = [Normality ORIGINAL group, 10, FOR EVERY x,y,z in0..n-1]

After

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 1000, Yes]

Conditions: none

Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups.

Step identifiers and dependencies

Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002

S004 · implies

Before

source roles = [Normality ORIGINAL group, 3]

After

derived = [[6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [[0, 6, 3, 3], [1, 5, 3, 3], [2, 4, 3, 3], [3, 3, 3, 3], [4, 2, 3, 3], [5, 1, 3, 3], [6, 0, 3, 3], [7, 9, 3, 3], [8, 8, 3, 3], [9, 7, 3, 3]], Yes]

Conditions: none

Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises.

Step identifiers and dependencies

Rule: Group_ALL_two_sided_inverses · Uses: G001, S003

S005 · implies

Before

source roles = [Normality ORIGINAL H, [3], 3]

After

derived = [Yes, [[3, 3, 3, Yes]], Yes, [[3, 3, Yes]], Yes, Yes]

Conditions: none

H is nonempty and unique.Check identity membership,all products and every inverse separately;their conjunction qualifies a subgroup.Size divisibility or finiteness alone is insufficient.

Step identifiers and dependencies

Rule: Subgroup_identity_ALL_products_ALL_inverses_separate · Uses: G001, S004

S006 · implies

Before

source roles = [Normality ORIGINAL H, 0, [3], 6]

After

derived = [[[3, 0, 6, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S005

S007 · implies

Before

source roles = [Normality ORIGINAL H, 1, [3], 5]

After

derived = [[[3, 1, 5, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S006

S008 · implies

Before

source roles = [Normality ORIGINAL H, 2, [3], 4]

After

derived = [[[3, 2, 4, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S007

S009 · implies

Before

source roles = [Normality ORIGINAL H, 3, [3], 3]

After

derived = [[[3, 3, 3, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S008

S010 · implies

Before

source roles = [Normality ORIGINAL H, 4, [3], 2]

After

derived = [[[3, 4, 2, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S009

S011 · implies

Before

source roles = [Normality ORIGINAL H, 5, [3], 1]

After

derived = [[[3, 5, 1, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S010

S012 · implies

Before

source roles = [Normality ORIGINAL H, 6, [3], 0]

After

derived = [[[3, 6, 0, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S011

S013 · implies

Before

source roles = [Normality ORIGINAL H, 7, [3], 9]

After

derived = [[[3, 7, 9, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S012

S014 · implies

Before

source roles = [Normality ORIGINAL H, 8, [3], 8]

After

derived = [[[3, 8, 8, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S013

S015 · implies

Before

source roles = [Normality ORIGINAL H, 9, [3], 7]

After

derived = [[[3, 9, 7, 3]], [3], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S014

S016 · implies

Before

source roles = [0, [3]]

After

derived = [[0], [0], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S015

S017 · implies

Before

source roles = [1, [3]]

After

derived = [[1], [1], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S016

S018 · implies

Before

source roles = [2, [3]]

After

derived = [[2], [2], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S017

S019 · implies

Before

source roles = [3, [3]]

After

derived = [[3], [3], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S018

S020 · implies

Before

source roles = [4, [3]]

After

derived = [[4], [4], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S019

S021 · implies

Before

source roles = [5, [3]]

After

derived = [[5], [5], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S020

S022 · implies

Before

source roles = [6, [3]]

After

derived = [[6], [6], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S021

S023 · implies

Before

source roles = [7, [3]]

After

derived = [[7], [7], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S022

S024 · implies

Before

source roles = [8, [3]]

After

derived = [[8], [8], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S023

S025 · implies

Before

source roles = [9, [3]]

After

derived = [[9], [9], Yes]

Conditions: none

These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group.

Step identifiers and dependencies

Rule: Normality_ALL_g_LEFT_RIGHT_cosets · Uses: G001, S024

S026 · implies

Before

source roles = [[3], Yes, Yes, Yes]

After

derived = [gHg^-1=H iff gH=Hg for EVERY g]

Conditions: none

Right multiplication by g is a bijection,turning gHg^-1=H into gH=Hg;right multiply by g inverse for the converse.No quotient multiplication is claimed here.

Step identifiers and dependencies

Rule: Normality_GENERAL_conjugates_iff_left_right_cosets · Uses: G001, S025

Reverse checks

  • 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

([3], [[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], [8, [8], [8], Yes], [9, [9], [9], Yes]], Yes, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)

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: 4967f171513391569e09eb77ed94cfd7117029e4b0d96afc542bb53aa2530f99
  • candidate record SHA-256: a2ab7897963c179e5dac3404e6da867ec1aac100542359383d5ee008746e455b
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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