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Question 24826: A Conjugacy Class Is Every Image of One Element under Conjugation

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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R0048-24826 · q-20261015-advance-20261114-r03-finite-group-conjugacy-class-0026

Original question and givens

Use the complete finite group table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]] with identity=0. First certify all group axioms and inverses, then form the full left conjugation action g*x=g*x*g^(-1). Every acting g and acted-on element is included, not a sample. Specified element=3. Give (element, action_image_rows, conjugacy_class, certificate).

  • G001 — Use the complete finite group table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]] with identity=0. First certify all group axioms and inverses, then form the full left conjugation action g*x=g*x*g^(-1). Every acting g and acted-on element is included, not a sample. Specified element=3. Give (element, action_image_rows, conjugacy_class, 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 = [conjugacy-orbit-class]

After

derived = [[element, action_image_rows, conjugacy_class, 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 = [Conjugacy ORIGINAL group, 64, [[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]], 0]

After

derived = [6, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5]], 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 = [Conjugacy ORIGINAL group, 6, 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]], 216, 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 = [Conjugacy ORIGINAL group, 0]

After

derived = [[0, 2, 1, 3, 4, 5], [[0, 0, 0, 0], [1, 2, 0, 0], [2, 1, 0, 0], [3, 3, 0, 0], [4, 4, 0, 0], [5, 5, 0, 0]], 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 = [0, 0, [0, 1, 2, 3, 4, 5]]

After

derived = [[0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]]

Conditions: none

For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup.

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S004

S006 · implies

Before

source roles = [1, 2, [0, 1, 2, 3, 4, 5]]

After

derived = [[1, 2, 0, 4, 5, 3], [0, 1, 2, 5, 3, 4]]

Conditions: none

For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup.

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S005

S007 · implies

Before

source roles = [2, 1, [0, 1, 2, 3, 4, 5]]

After

derived = [[2, 0, 1, 5, 3, 4], [0, 1, 2, 4, 5, 3]]

Conditions: none

For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup.

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S006

S008 · implies

Before

source roles = [3, 3, [0, 1, 2, 3, 4, 5]]

After

derived = [[3, 5, 4, 0, 2, 1], [0, 2, 1, 3, 5, 4]]

Conditions: none

For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup.

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S007

S009 · implies

Before

source roles = [4, 4, [0, 1, 2, 3, 4, 5]]

After

derived = [[4, 3, 5, 1, 0, 2], [0, 2, 1, 5, 4, 3]]

Conditions: none

For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup.

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S008

S010 · implies

Before

source roles = [5, 5, [0, 1, 2, 3, 4, 5]]

After

derived = [[5, 4, 3, 2, 1, 0], [0, 2, 1, 4, 3, 5]]

Conditions: none

For every original element x,compute gx then multiply by g inverse to form the complete acting row.This is an element action,not a conjugated subgroup.

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S009

S011 · implies

Before

source roles = [0, 6, [[0, 1, 2, 3, 4, 5], [0, 1, 2, 5, 3, 4], [0, 1, 2, 4, 5, 3], [0, 2, 1, 3, 5, 4], [0, 2, 1, 5, 4, 3], [0, 2, 1, 4, 3, 5]]]

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, (gh)x(gh)^-1=g(hxh^-1)g^-1]

Conditions: none

Check composition for every g,h,x.The same formula follows from(gh)^-1=h^-1g^-1 and associativity;check identity separately.Listing an action table alone would not prove its axioms.

Step identifiers and dependencies

Rule: Conjugacy_ALL_action_axioms_and_GENERAL_algebra · Uses: G001, S010

S012 · implies

Before

source roles = [[[0, 1, 2, 3, 4, 5], [0, 1, 2, 5, 3, 4], [0, 1, 2, 4, 5, 3], [0, 2, 1, 3, 5, 4], [0, 2, 1, 5, 4, 3], [0, 2, 1, 4, 3, 5]]]

After

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

Conditions: none

Identity gives reflexivity,inverses symmetry and products transitivity,so complete orbits are equal or disjoint and cover G.Source family sorting does not make classes cosets;no centralizer/class-size formula or class equation is introduced.

Step identifiers and dependencies

Rule: Conjugacy_complete_orbits_equal_or_disjoint_partition · Uses: G001, S011

S013 · implies

Before

source roles = [3]

After

derived = [[[0, 3], [1, 5], [2, 4], [3, 3], [4, 5], [5, 4]], [3, 4, 5]]

Conditions: none

Retain the image for every acting g,then deduplicate into the full sorted class.

Step identifiers and dependencies

Rule: Conjugacy_actual_element_ALL_images_not_coset · Uses: G001, S012

Reverse checks

  • 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, 3], [1, 5], [2, 4], [3, 3], [4, 5], [5, 4]], [3, 4, 5], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)

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: a2003501d8b455be2293f2b22a93fb1d9957bee563a5103a167120daef7cc88d
  • candidate record SHA-256: 9ceade0c1817fe335f693f2a6d05ead817581b53d8d337943aa2b5516979a8b7
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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