Complete Conjugation Actions, Classes and Nonconjugacy Witnesses
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261114-r03-finite-group-conjugacy-class-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. The Entire Conjugation Table Must Satisfy Both Action Axioms
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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[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]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[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]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
2. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 5, 6, 7, 4], [2, 3, 0, 1, 6, 7, 4, 5], [3, 0, 1, 2, 7, 4, 5, 6], [4, 7, 6, 5, 0, 3, 2, 1], [5, 4, 7, 6, 1, 0, 3, 2], [6, 5, 4, 7, 2, 1, 0, 3], [7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 6, 7, 4, 5], [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 6, 7, 4, 5], [0, 3, 2, 1, 4, 7, 6, 5], [0, 3, 2, 1, 6, 5, 4, 7], [0, 3, 2, 1, 4, 7, 6, 5], [0, 3, 2, 1, 6, 5, 4, 7]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 6, 7, 4, 5], [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 6, 7, 4, 5], [0, 3, 2, 1, 4, 7, 6, 5], [0, 3, 2, 1, 6, 5, 4, 7], [0, 3, 2, 1, 4, 7, 6, 5], [0, 3, 2, 1, 6, 5, 4, 7]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
3. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 7, 8, 9, 5, 6], [0, 1, 2, 3, 4, 9, 5, 6, 7, 8], [0, 1, 2, 3, 4, 6, 7, 8, 9, 5], [0, 1, 2, 3, 4, 8, 9, 5, 6, 7], [0, 4, 3, 2, 1, 5, 9, 8, 7, 6], [0, 4, 3, 2, 1, 7, 6, 5, 9, 8], [0, 4, 3, 2, 1, 9, 8, 7, 6, 5], [0, 4, 3, 2, 1, 6, 5, 9, 8, 7], [0, 4, 3, 2, 1, 8, 7, 6, 5, 9]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 4, 7, 8, 9, 5, 6], [0, 1, 2, 3, 4, 9, 5, 6, 7, 8], [0, 1, 2, 3, 4, 6, 7, 8, 9, 5], [0, 1, 2, 3, 4, 8, 9, 5, 6, 7], [0, 4, 3, 2, 1, 5, 9, 8, 7, 6], [0, 4, 3, 2, 1, 7, 6, 5, 9, 8], [0, 4, 3, 2, 1, 9, 8, 7, 6, 5], [0, 4, 3, 2, 1, 6, 5, 9, 8, 7], [0, 4, 3, 2, 1, 8, 7, 6, 5, 9]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
4. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 6, 7], [0, 1, 2, 3, 4, 5, 10, 11, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 6, 7], [0, 1, 2, 3, 4, 5, 10, 11, 6, 7, 8, 9], [0, 5, 4, 3, 2, 1, 6, 11, 10, 9, 8, 7], [0, 5, 4, 3, 2, 1, 8, 7, 6, 11, 10, 9], [0, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 11], [0, 5, 4, 3, 2, 1, 6, 11, 10, 9, 8, 7], [0, 5, 4, 3, 2, 1, 8, 7, 6, 11, 10, 9], [0, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 11]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 6, 7], [0, 1, 2, 3, 4, 5, 10, 11, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 6, 7], [0, 1, 2, 3, 4, 5, 10, 11, 6, 7, 8, 9], [0, 5, 4, 3, 2, 1, 6, 11, 10, 9, 8, 7], [0, 5, 4, 3, 2, 1, 8, 7, 6, 11, 10, 9], [0, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 11], [0, 5, 4, 3, 2, 1, 6, 11, 10, 9, 8, 7], [0, 5, 4, 3, 2, 1, 8, 7, 6, 11, 10, 9], [0, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 11]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
5. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 7], [2, 3, 4, 5, 6, 0, 1, 9, 10, 11, 12, 13, 7, 8], [3, 4, 5, 6, 0, 1, 2, 10, 11, 12, 13, 7, 8, 9], [4, 5, 6, 0, 1, 2, 3, 11, 12, 13, 7, 8, 9, 10], [5, 6, 0, 1, 2, 3, 4, 12, 13, 7, 8, 9, 10, 11], [6, 0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12], [7, 13, 12, 11, 10, 9, 8, 0, 6, 5, 4, 3, 2, 1], [8, 7, 13, 12, 11, 10, 9, 1, 0, 6, 5, 4, 3, 2], [9, 8, 7, 13, 12, 11, 10, 2, 1, 0, 6, 5, 4, 3], [10, 9, 8, 7, 13, 12, 11, 3, 2, 1, 0, 6, 5, 4], [11, 10, 9, 8, 7, 13, 12, 4, 3, 2, 1, 0, 6, 5], [12, 11, 10, 9, 8, 7, 13, 5, 4, 3, 2, 1, 0, 6], [13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 7, 8], [0, 1, 2, 3, 4, 5, 6, 11, 12, 13, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 13, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 7], [0, 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 12, 13, 7, 8, 9, 10, 11], [0, 6, 5, 4, 3, 2, 1, 7, 13, 12, 11, 10, 9, 8], [0, 6, 5, 4, 3, 2, 1, 9, 8, 7, 13, 12, 11, 10], [0, 6, 5, 4, 3, 2, 1, 11, 10, 9, 8, 7, 13, 12], [0, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 8, 7], [0, 6, 5, 4, 3, 2, 1, 8, 7, 13, 12, 11, 10, 9], [0, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 13, 12, 11], [0, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 13]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 7, 8], [0, 1, 2, 3, 4, 5, 6, 11, 12, 13, 7, 8, 9, 10], [0, 1, 2, 3, 4, 5, 6, 13, 7, 8, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 7], [0, 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 7, 8, 9], [0, 1, 2, 3, 4, 5, 6, 12, 13, 7, 8, 9, 10, 11], [0, 6, 5, 4, 3, 2, 1, 7, 13, 12, 11, 10, 9, 8], [0, 6, 5, 4, 3, 2, 1, 9, 8, 7, 13, 12, 11, 10], [0, 6, 5, 4, 3, 2, 1, 11, 10, 9, 8, 7, 13, 12], [0, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 8, 7], [0, 6, 5, 4, 3, 2, 1, 8, 7, 13, 12, 11, 10, 9], [0, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 13, 12, 11], [0, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 13]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
6. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 15, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 14, 15, 8, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 15, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 14, 15, 8, 9, 10, 11, 12, 13], [0, 7, 6, 5, 4, 3, 2, 1, 8, 15, 14, 13, 12, 11, 10, 9], [0, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 15, 14, 13, 12, 11], [0, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 15, 14, 13], [0, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 8, 15], [0, 7, 6, 5, 4, 3, 2, 1, 8, 15, 14, 13, 12, 11, 10, 9], [0, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 15, 14, 13, 12, 11], [0, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 15, 14, 13], [0, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 8, 15]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 15, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 14, 15, 8, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 8, 9], [0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 15, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 14, 15, 8, 9, 10, 11, 12, 13], [0, 7, 6, 5, 4, 3, 2, 1, 8, 15, 14, 13, 12, 11, 10, 9], [0, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 15, 14, 13, 12, 11], [0, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 15, 14, 13], [0, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 8, 15], [0, 7, 6, 5, 4, 3, 2, 1, 8, 15, 14, 13, 12, 11, 10, 9], [0, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 15, 14, 13, 12, 11], [0, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 15, 14, 13], [0, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 8, 15]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
7. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 15, 16, 17, 9, 10, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 16, 17, 9, 10, 11, 12, 13, 14, 15], [0, 8, 7, 6, 5, 4, 3, 2, 1, 9, 17, 16, 15, 14, 13, 12, 11, 10], [0, 8, 7, 6, 5, 4, 3, 2, 1, 11, 10, 9, 17, 16, 15, 14, 13, 12], [0, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 17, 16, 15, 14], [0, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 10, 9, 17, 16], [0, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 10, 9], [0, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 17, 16, 15, 14, 13, 12, 11], [0, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 17, 16, 15, 14, 13], [0, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 17, 16, 15], [0, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 17]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 9, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 9, 10, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 15, 16, 17, 9, 10, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 9], [0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17, 9, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 16, 17, 9, 10, 11, 12, 13, 14, 15], [0, 8, 7, 6, 5, 4, 3, 2, 1, 9, 17, 16, 15, 14, 13, 12, 11, 10], [0, 8, 7, 6, 5, 4, 3, 2, 1, 11, 10, 9, 17, 16, 15, 14, 13, 12], [0, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 17, 16, 15, 14], [0, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 10, 9, 17, 16], [0, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 10, 9], [0, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 17, 16, 15, 14, 13, 12, 11], [0, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 17, 16, 15, 14, 13], [0, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 17, 16, 15], [0, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 17]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
8. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 19, 18, 17, 16, 15, 14, 13], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 19, 18, 17, 16, 15], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 19, 18, 17], [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 19]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
9. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19], [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
10. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3], [15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4], [16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5], [17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6], [18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7], [19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8], [20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21], [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
11. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4], [17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5], [18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6], [19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7], [20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8], [21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23], [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
12. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5], [19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6], [20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7], [21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8], [22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25], [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
13. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6], [21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7], [22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8], [23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27], [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
14. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7], [23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
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([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29], [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
15. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8], [25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31], [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
16. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33], [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
17. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35], [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
18. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37], [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
19. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39], [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
20. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41], [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
21. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43], [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
22. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45], [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
23. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
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([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47], [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
24. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49], [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
25. The Entire Conjugation Table Must Satisfy Both Action Axioms
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25], [52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26], [53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (conjugation_action, identity_action, composition_action, certificate).
View answer and explanation
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 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37, 36, 35, 34, 33, 32, 31, 30], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action)
Give the full action table in group-index order and separately verify identity action and composition action. This is conjugation on elements, not a subgroup conjugate set or coset table. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51], [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53]], Yes, Yes, g_dot_x_equals_g_x_g_inverse_is_a_complete_left_action).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-action-table · candidate
26. A Conjugacy Class Is Every Image of One Element under Conjugation
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).
View answer and explanation
(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)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (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).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
27. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 5, 6, 7, 4], [2, 3, 0, 1, 6, 7, 4, 5], [3, 0, 1, 2, 7, 4, 5, 6], [4, 7, 6, 5, 0, 3, 2, 1], [5, 4, 7, 6, 1, 0, 3, 2], [6, 5, 4, 7, 2, 1, 0, 3], [7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 3], [5, 3], [6, 3], [7, 3]], [1, 3], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 3], [5, 3], [6, 3], [7, 3]], [1, 3], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
28. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 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=5. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(5, [[0, 5], [1, 7], [2, 9], [3, 6], [4, 8], [5, 5], [6, 7], [7, 9], [8, 6], [9, 8]], [5, 6, 7, 8, 9], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (5, [[0, 5], [1, 7], [2, 9], [3, 6], [4, 8], [5, 5], [6, 7], [7, 9], [8, 6], [9, 8]], [5, 6, 7, 8, 9], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
29. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 5], [7, 5], [8, 5], [9, 5], [10, 5], [11, 5]], [1, 5], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 5], [7, 5], [8, 5], [9, 5], [10, 5], [11, 5]], [1, 5], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
30. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 7], [2, 3, 4, 5, 6, 0, 1, 9, 10, 11, 12, 13, 7, 8], [3, 4, 5, 6, 0, 1, 2, 10, 11, 12, 13, 7, 8, 9], [4, 5, 6, 0, 1, 2, 3, 11, 12, 13, 7, 8, 9, 10], [5, 6, 0, 1, 2, 3, 4, 12, 13, 7, 8, 9, 10, 11], [6, 0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12], [7, 13, 12, 11, 10, 9, 8, 0, 6, 5, 4, 3, 2, 1], [8, 7, 13, 12, 11, 10, 9, 1, 0, 6, 5, 4, 3, 2], [9, 8, 7, 13, 12, 11, 10, 2, 1, 0, 6, 5, 4, 3], [10, 9, 8, 7, 13, 12, 11, 3, 2, 1, 0, 6, 5, 4], [11, 10, 9, 8, 7, 13, 12, 4, 3, 2, 1, 0, 6, 5], [12, 11, 10, 9, 8, 7, 13, 5, 4, 3, 2, 1, 0, 6], [13, 12, 11, 10, 9, 8, 7, 6, 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=7. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(7, [[0, 7], [1, 9], [2, 11], [3, 13], [4, 8], [5, 10], [6, 12], [7, 7], [8, 9], [9, 11], [10, 13], [11, 8], [12, 10], [13, 12]], [7, 8, 9, 10, 11, 12, 13], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (7, [[0, 7], [1, 9], [2, 11], [3, 13], [4, 8], [5, 10], [6, 12], [7, 7], [8, 9], [9, 11], [10, 13], [11, 8], [12, 10], [13, 12]], [7, 8, 9, 10, 11, 12, 13], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
31. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 7], [9, 7], [10, 7], [11, 7], [12, 7], [13, 7], [14, 7], [15, 7]], [1, 7], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 7], [9, 7], [10, 7], [11, 7], [12, 7], [13, 7], [14, 7], [15, 7]], [1, 7], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
32. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=9. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(9, [[0, 9], [1, 11], [2, 13], [3, 15], [4, 17], [5, 10], [6, 12], [7, 14], [8, 16], [9, 9], [10, 11], [11, 13], [12, 15], [13, 17], [14, 10], [15, 12], [16, 14], [17, 16]], [9, 10, 11, 12, 13, 14, 15, 16, 17], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (9, [[0, 9], [1, 11], [2, 13], [3, 15], [4, 17], [5, 10], [6, 12], [7, 14], [8, 16], [9, 9], [10, 11], [11, 13], [12, 15], [13, 17], [14, 10], [15, 12], [16, 14], [17, 16]], [9, 10, 11, 12, 13, 14, 15, 16, 17], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
33. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 9], [11, 9], [12, 9], [13, 9], [14, 9], [15, 9], [16, 9], [17, 9], [18, 9], [19, 9]], [1, 9], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 9], [11, 9], [12, 9], [13, 9], [14, 9], [15, 9], [16, 9], [17, 9], [18, 9], [19, 9]], [1, 9], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
34. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=11. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(11, [[0, 11], [1, 13], [2, 15], [3, 17], [4, 19], [5, 21], [6, 12], [7, 14], [8, 16], [9, 18], [10, 20], [11, 11], [12, 13], [13, 15], [14, 17], [15, 19], [16, 21], [17, 12], [18, 14], [19, 16], [20, 18], [21, 20]], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (11, [[0, 11], [1, 13], [2, 15], [3, 17], [4, 19], [5, 21], [6, 12], [7, 14], [8, 16], [9, 18], [10, 20], [11, 11], [12, 13], [13, 15], [14, 17], [15, 19], [16, 21], [17, 12], [18, 14], [19, 16], [20, 18], [21, 20]], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
35. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3], [15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4], [16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5], [17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6], [18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7], [19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8], [20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 11], [13, 11], [14, 11], [15, 11], [16, 11], [17, 11], [18, 11], [19, 11], [20, 11], [21, 11], [22, 11], [23, 11]], [1, 11], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 11], [13, 11], [14, 11], [15, 11], [16, 11], [17, 11], [18, 11], [19, 11], [20, 11], [21, 11], [22, 11], [23, 11]], [1, 11], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
36. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4], [17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5], [18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6], [19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7], [20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8], [21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=13. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(13, [[0, 13], [1, 15], [2, 17], [3, 19], [4, 21], [5, 23], [6, 25], [7, 14], [8, 16], [9, 18], [10, 20], [11, 22], [12, 24], [13, 13], [14, 15], [15, 17], [16, 19], [17, 21], [18, 23], [19, 25], [20, 14], [21, 16], [22, 18], [23, 20], [24, 22], [25, 24]], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (13, [[0, 13], [1, 15], [2, 17], [3, 19], [4, 21], [5, 23], [6, 25], [7, 14], [8, 16], [9, 18], [10, 20], [11, 22], [12, 24], [13, 13], [14, 15], [15, 17], [16, 19], [17, 21], [18, 23], [19, 25], [20, 14], [21, 16], [22, 18], [23, 20], [24, 22], [25, 24]], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
37. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5], [19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6], [20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7], [21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8], [22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 13], [15, 13], [16, 13], [17, 13], [18, 13], [19, 13], [20, 13], [21, 13], [22, 13], [23, 13], [24, 13], [25, 13], [26, 13], [27, 13]], [1, 13], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 13], [15, 13], [16, 13], [17, 13], [18, 13], [19, 13], [20, 13], [21, 13], [22, 13], [23, 13], [24, 13], [25, 13], [26, 13], [27, 13]], [1, 13], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
38. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6], [21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7], [22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8], [23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=15. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(15, [[0, 15], [1, 17], [2, 19], [3, 21], [4, 23], [5, 25], [6, 27], [7, 29], [8, 16], [9, 18], [10, 20], [11, 22], [12, 24], [13, 26], [14, 28], [15, 15], [16, 17], [17, 19], [18, 21], [19, 23], [20, 25], [21, 27], [22, 29], [23, 16], [24, 18], [25, 20], [26, 22], [27, 24], [28, 26], [29, 28]], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (15, [[0, 15], [1, 17], [2, 19], [3, 21], [4, 23], [5, 25], [6, 27], [7, 29], [8, 16], [9, 18], [10, 20], [11, 22], [12, 24], [13, 26], [14, 28], [15, 15], [16, 17], [17, 19], [18, 21], [19, 23], [20, 25], [21, 27], [22, 29], [23, 16], [24, 18], [25, 20], [26, 22], [27, 24], [28, 26], [29, 28]], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
39. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7], [23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 15], [17, 15], [18, 15], [19, 15], [20, 15], [21, 15], [22, 15], [23, 15], [24, 15], [25, 15], [26, 15], [27, 15], [28, 15], [29, 15], [30, 15], [31, 15]], [1, 15], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 15], [17, 15], [18, 15], [19, 15], [20, 15], [21, 15], [22, 15], [23, 15], [24, 15], [25, 15], [26, 15], [27, 15], [28, 15], [29, 15], [30, 15], [31, 15]], [1, 15], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
40. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8], [25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=17. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(17, [[0, 17], [1, 19], [2, 21], [3, 23], [4, 25], [5, 27], [6, 29], [7, 31], [8, 33], [9, 18], [10, 20], [11, 22], [12, 24], [13, 26], [14, 28], [15, 30], [16, 32], [17, 17], [18, 19], [19, 21], [20, 23], [21, 25], [22, 27], [23, 29], [24, 31], [25, 33], [26, 18], [27, 20], [28, 22], [29, 24], [30, 26], [31, 28], [32, 30], [33, 32]], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (17, [[0, 17], [1, 19], [2, 21], [3, 23], [4, 25], [5, 27], [6, 29], [7, 31], [8, 33], [9, 18], [10, 20], [11, 22], [12, 24], [13, 26], [14, 28], [15, 30], [16, 32], [17, 17], [18, 19], [19, 21], [20, 23], [21, 25], [22, 27], [23, 29], [24, 31], [25, 33], [26, 18], [27, 20], [28, 22], [29, 24], [30, 26], [31, 28], [32, 30], [33, 32]], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
41. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 17], [19, 17], [20, 17], [21, 17], [22, 17], [23, 17], [24, 17], [25, 17], [26, 17], [27, 17], [28, 17], [29, 17], [30, 17], [31, 17], [32, 17], [33, 17], [34, 17], [35, 17]], [1, 17], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 17], [19, 17], [20, 17], [21, 17], [22, 17], [23, 17], [24, 17], [25, 17], [26, 17], [27, 17], [28, 17], [29, 17], [30, 17], [31, 17], [32, 17], [33, 17], [34, 17], [35, 17]], [1, 17], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
42. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=19. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(19, [[0, 19], [1, 21], [2, 23], [3, 25], [4, 27], [5, 29], [6, 31], [7, 33], [8, 35], [9, 37], [10, 20], [11, 22], [12, 24], [13, 26], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 19], [20, 21], [21, 23], [22, 25], [23, 27], [24, 29], [25, 31], [26, 33], [27, 35], [28, 37], [29, 20], [30, 22], [31, 24], [32, 26], [33, 28], [34, 30], [35, 32], [36, 34], [37, 36]], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (19, [[0, 19], [1, 21], [2, 23], [3, 25], [4, 27], [5, 29], [6, 31], [7, 33], [8, 35], [9, 37], [10, 20], [11, 22], [12, 24], [13, 26], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 19], [20, 21], [21, 23], [22, 25], [23, 27], [24, 29], [25, 31], [26, 33], [27, 35], [28, 37], [29, 20], [30, 22], [31, 24], [32, 26], [33, 28], [34, 30], [35, 32], [36, 34], [37, 36]], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
43. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 19], [21, 19], [22, 19], [23, 19], [24, 19], [25, 19], [26, 19], [27, 19], [28, 19], [29, 19], [30, 19], [31, 19], [32, 19], [33, 19], [34, 19], [35, 19], [36, 19], [37, 19], [38, 19], [39, 19]], [1, 19], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 19], [21, 19], [22, 19], [23, 19], [24, 19], [25, 19], [26, 19], [27, 19], [28, 19], [29, 19], [30, 19], [31, 19], [32, 19], [33, 19], [34, 19], [35, 19], [36, 19], [37, 19], [38, 19], [39, 19]], [1, 19], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
44. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=21. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(21, [[0, 21], [1, 23], [2, 25], [3, 27], [4, 29], [5, 31], [6, 33], [7, 35], [8, 37], [9, 39], [10, 41], [11, 22], [12, 24], [13, 26], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 38], [20, 40], [21, 21], [22, 23], [23, 25], [24, 27], [25, 29], [26, 31], [27, 33], [28, 35], [29, 37], [30, 39], [31, 41], [32, 22], [33, 24], [34, 26], [35, 28], [36, 30], [37, 32], [38, 34], [39, 36], [40, 38], [41, 40]], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (21, [[0, 21], [1, 23], [2, 25], [3, 27], [4, 29], [5, 31], [6, 33], [7, 35], [8, 37], [9, 39], [10, 41], [11, 22], [12, 24], [13, 26], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 38], [20, 40], [21, 21], [22, 23], [23, 25], [24, 27], [25, 29], [26, 31], [27, 33], [28, 35], [29, 37], [30, 39], [31, 41], [32, 22], [33, 24], [34, 26], [35, 28], [36, 30], [37, 32], [38, 34], [39, 36], [40, 38], [41, 40]], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
45. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 21], [23, 21], [24, 21], [25, 21], [26, 21], [27, 21], [28, 21], [29, 21], [30, 21], [31, 21], [32, 21], [33, 21], [34, 21], [35, 21], [36, 21], [37, 21], [38, 21], [39, 21], [40, 21], [41, 21], [42, 21], [43, 21]], [1, 21], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 21], [23, 21], [24, 21], [25, 21], [26, 21], [27, 21], [28, 21], [29, 21], [30, 21], [31, 21], [32, 21], [33, 21], [34, 21], [35, 21], [36, 21], [37, 21], [38, 21], [39, 21], [40, 21], [41, 21], [42, 21], [43, 21]], [1, 21], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
46. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=23. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(23, [[0, 23], [1, 25], [2, 27], [3, 29], [4, 31], [5, 33], [6, 35], [7, 37], [8, 39], [9, 41], [10, 43], [11, 45], [12, 24], [13, 26], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 38], [20, 40], [21, 42], [22, 44], [23, 23], [24, 25], [25, 27], [26, 29], [27, 31], [28, 33], [29, 35], [30, 37], [31, 39], [32, 41], [33, 43], [34, 45], [35, 24], [36, 26], [37, 28], [38, 30], [39, 32], [40, 34], [41, 36], [42, 38], [43, 40], [44, 42], [45, 44]], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (23, [[0, 23], [1, 25], [2, 27], [3, 29], [4, 31], [5, 33], [6, 35], [7, 37], [8, 39], [9, 41], [10, 43], [11, 45], [12, 24], [13, 26], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 38], [20, 40], [21, 42], [22, 44], [23, 23], [24, 25], [25, 27], [26, 29], [27, 31], [28, 33], [29, 35], [30, 37], [31, 39], [32, 41], [33, 43], [34, 45], [35, 24], [36, 26], [37, 28], [38, 30], [39, 32], [40, 34], [41, 36], [42, 38], [43, 40], [44, 42], [45, 44]], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
47. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 23], [25, 23], [26, 23], [27, 23], [28, 23], [29, 23], [30, 23], [31, 23], [32, 23], [33, 23], [34, 23], [35, 23], [36, 23], [37, 23], [38, 23], [39, 23], [40, 23], [41, 23], [42, 23], [43, 23], [44, 23], [45, 23], [46, 23], [47, 23]], [1, 23], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 23], [25, 23], [26, 23], [27, 23], [28, 23], [29, 23], [30, 23], [31, 23], [32, 23], [33, 23], [34, 23], [35, 23], [36, 23], [37, 23], [38, 23], [39, 23], [40, 23], [41, 23], [42, 23], [43, 23], [44, 23], [45, 23], [46, 23], [47, 23]], [1, 23], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
48. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=25. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(25, [[0, 25], [1, 27], [2, 29], [3, 31], [4, 33], [5, 35], [6, 37], [7, 39], [8, 41], [9, 43], [10, 45], [11, 47], [12, 49], [13, 26], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 38], [20, 40], [21, 42], [22, 44], [23, 46], [24, 48], [25, 25], [26, 27], [27, 29], [28, 31], [29, 33], [30, 35], [31, 37], [32, 39], [33, 41], [34, 43], [35, 45], [36, 47], [37, 49], [38, 26], [39, 28], [40, 30], [41, 32], [42, 34], [43, 36], [44, 38], [45, 40], [46, 42], [47, 44], [48, 46], [49, 48]], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (25, [[0, 25], [1, 27], [2, 29], [3, 31], [4, 33], [5, 35], [6, 37], [7, 39], [8, 41], [9, 43], [10, 45], [11, 47], [12, 49], [13, 26], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 38], [20, 40], [21, 42], [22, 44], [23, 46], [24, 48], [25, 25], [26, 27], [27, 29], [28, 31], [29, 33], [30, 35], [31, 37], [32, 39], [33, 41], [34, 43], [35, 45], [36, 47], [37, 49], [38, 26], [39, 28], [40, 30], [41, 32], [42, 34], [43, 36], [44, 38], [45, 40], [46, 42], [47, 44], [48, 46], [49, 48]], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
49. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=1. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, 25], [27, 25], [28, 25], [29, 25], [30, 25], [31, 25], [32, 25], [33, 25], [34, 25], [35, 25], [36, 25], [37, 25], [38, 25], [39, 25], [40, 25], [41, 25], [42, 25], [43, 25], [44, 25], [45, 25], [46, 25], [47, 25], [48, 25], [49, 25], [50, 25], [51, 25]], [1, 25], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, 25], [27, 25], [28, 25], [29, 25], [30, 25], [31, 25], [32, 25], [33, 25], [34, 25], [35, 25], [36, 25], [37, 25], [38, 25], [39, 25], [40, 25], [41, 25], [42, 25], [43, 25], [44, 25], [45, 25], [46, 25], [47, 25], [48, 25], [49, 25], [50, 25], [51, 25]], [1, 25], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
50. A Conjugacy Class Is Every Image of One Element under Conjugation
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25], [52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26], [53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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=27. Give (element, action_image_rows, conjugacy_class, certificate).
View answer and explanation
(27, [[0, 27], [1, 29], [2, 31], [3, 33], [4, 35], [5, 37], [6, 39], [7, 41], [8, 43], [9, 45], [10, 47], [11, 49], [12, 51], [13, 53], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 38], [20, 40], [21, 42], [22, 44], [23, 46], [24, 48], [25, 50], [26, 52], [27, 27], [28, 29], [29, 31], [30, 33], [31, 35], [32, 37], [33, 39], [34, 41], [35, 43], [36, 45], [37, 47], [38, 49], [39, 51], [40, 53], [41, 28], [42, 30], [43, 32], [44, 34], [45, 36], [46, 38], [47, 40], [48, 42], [49, 44], [50, 46], [51, 48], [52, 50], [53, 52]], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset)
List every [g,g*x*g^(-1)] and its sorted distinct orbit. Do not sample acting elements or replace the orbit by a left coset. No centralizer/class-size formula is used. The exact result, retaining the requested order, is (27, [[0, 27], [1, 29], [2, 31], [3, 33], [4, 35], [5, 37], [6, 39], [7, 41], [8, 43], [9, 45], [10, 47], [11, 49], [12, 51], [13, 53], [14, 28], [15, 30], [16, 32], [17, 34], [18, 36], [19, 38], [20, 40], [21, 42], [22, 44], [23, 46], [24, 48], [25, 50], [26, 52], [27, 27], [28, 29], [29, 31], [30, 33], [31, 35], [32, 37], [33, 39], [34, 41], [35, 43], [36, 45], [37, 47], [38, 49], [39, 51], [40, 53], [41, 28], [42, 30], [43, 32], [44, 34], [45, 36], [46, 38], [47, 40], [48, 42], [49, 44], [50, 46], [51, 48], [52, 50], [53, 52]], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], conjugacy_class_is_the_complete_conjugation_orbit_not_a_coset).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-orbit-class · candidate
51. All Conjugacy Classes Form a Complete Partition of the Group
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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 2], [3, 4, 5]], Yes, 6, 6, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 2], [3, 4, 5]], Yes, 6, 6, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
52. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 5, 6, 7, 4], [2, 3, 0, 1, 6, 7, 4, 5], [3, 0, 1, 2, 7, 4, 5, 6], [4, 7, 6, 5, 0, 3, 2, 1], [5, 4, 7, 6, 1, 0, 3, 2], [6, 5, 4, 7, 2, 1, 0, 3], [7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 3], [2], [4, 6], [5, 7]], Yes, 8, 8, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 3], [2], [4, 6], [5, 7]], Yes, 8, 8, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
53. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 4], [2, 3], [5, 6, 7, 8, 9]], Yes, 10, 10, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 4], [2, 3], [5, 6, 7, 8, 9]], Yes, 10, 10, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
54. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 5], [2, 4], [3], [6, 8, 10], [7, 9, 11]], Yes, 12, 12, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 5], [2, 4], [3], [6, 8, 10], [7, 9, 11]], Yes, 12, 12, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
55. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 7], [2, 3, 4, 5, 6, 0, 1, 9, 10, 11, 12, 13, 7, 8], [3, 4, 5, 6, 0, 1, 2, 10, 11, 12, 13, 7, 8, 9], [4, 5, 6, 0, 1, 2, 3, 11, 12, 13, 7, 8, 9, 10], [5, 6, 0, 1, 2, 3, 4, 12, 13, 7, 8, 9, 10, 11], [6, 0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12], [7, 13, 12, 11, 10, 9, 8, 0, 6, 5, 4, 3, 2, 1], [8, 7, 13, 12, 11, 10, 9, 1, 0, 6, 5, 4, 3, 2], [9, 8, 7, 13, 12, 11, 10, 2, 1, 0, 6, 5, 4, 3], [10, 9, 8, 7, 13, 12, 11, 3, 2, 1, 0, 6, 5, 4], [11, 10, 9, 8, 7, 13, 12, 4, 3, 2, 1, 0, 6, 5], [12, 11, 10, 9, 8, 7, 13, 5, 4, 3, 2, 1, 0, 6], [13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 6], [2, 5], [3, 4], [7, 8, 9, 10, 11, 12, 13]], Yes, 14, 14, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 6], [2, 5], [3, 4], [7, 8, 9, 10, 11, 12, 13]], Yes, 14, 14, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
56. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 7], [2, 6], [3, 5], [4], [8, 10, 12, 14], [9, 11, 13, 15]], Yes, 16, 16, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 7], [2, 6], [3, 5], [4], [8, 10, 12, 14], [9, 11, 13, 15]], Yes, 16, 16, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
57. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 8], [2, 7], [3, 6], [4, 5], [9, 10, 11, 12, 13, 14, 15, 16, 17]], Yes, 18, 18, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 8], [2, 7], [3, 6], [4, 5], [9, 10, 11, 12, 13, 14, 15, 16, 17]], Yes, 18, 18, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
58. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 9], [10, 12, 14, 16, 18], [11, 13, 15, 17, 19], [2, 8], [3, 7], [4, 6], [5]], Yes, 20, 20, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 9], [10, 12, 14, 16, 18], [11, 13, 15, 17, 19], [2, 8], [3, 7], [4, 6], [5]], Yes, 20, 20, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
59. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 10], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [2, 9], [3, 8], [4, 7], [5, 6]], Yes, 22, 22, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 10], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [2, 9], [3, 8], [4, 7], [5, 6]], Yes, 22, 22, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
60. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3], [15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4], [16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5], [17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6], [18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7], [19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8], [20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 11], [12, 14, 16, 18, 20, 22], [13, 15, 17, 19, 21, 23], [2, 10], [3, 9], [4, 8], [5, 7], [6]], Yes, 24, 24, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 11], [12, 14, 16, 18, 20, 22], [13, 15, 17, 19, 21, 23], [2, 10], [3, 9], [4, 8], [5, 7], [6]], Yes, 24, 24, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
61. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4], [17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5], [18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6], [19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7], [20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8], [21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 12], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [2, 11], [3, 10], [4, 9], [5, 8], [6, 7]], Yes, 26, 26, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 12], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [2, 11], [3, 10], [4, 9], [5, 8], [6, 7]], Yes, 26, 26, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
62. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5], [19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6], [20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7], [21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8], [22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 13], [14, 16, 18, 20, 22, 24, 26], [15, 17, 19, 21, 23, 25, 27], [2, 12], [3, 11], [4, 10], [5, 9], [6, 8], [7]], Yes, 28, 28, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 13], [14, 16, 18, 20, 22, 24, 26], [15, 17, 19, 21, 23, 25, 27], [2, 12], [3, 11], [4, 10], [5, 9], [6, 8], [7]], Yes, 28, 28, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
63. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6], [21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7], [22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8], [23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 14], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [2, 13], [3, 12], [4, 11], [5, 10], [6, 9], [7, 8]], Yes, 30, 30, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 14], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [2, 13], [3, 12], [4, 11], [5, 10], [6, 9], [7, 8]], Yes, 30, 30, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
64. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7], [23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 15], [16, 18, 20, 22, 24, 26, 28, 30], [17, 19, 21, 23, 25, 27, 29, 31], [2, 14], [3, 13], [4, 12], [5, 11], [6, 10], [7, 9], [8]], Yes, 32, 32, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 15], [16, 18, 20, 22, 24, 26, 28, 30], [17, 19, 21, 23, 25, 27, 29, 31], [2, 14], [3, 13], [4, 12], [5, 11], [6, 10], [7, 9], [8]], Yes, 32, 32, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
65. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8], [25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 16], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [2, 15], [3, 14], [4, 13], [5, 12], [6, 11], [7, 10], [8, 9]], Yes, 34, 34, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 16], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [2, 15], [3, 14], [4, 13], [5, 12], [6, 11], [7, 10], [8, 9]], Yes, 34, 34, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
66. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 17], [18, 20, 22, 24, 26, 28, 30, 32, 34], [19, 21, 23, 25, 27, 29, 31, 33, 35], [2, 16], [3, 15], [4, 14], [5, 13], [6, 12], [7, 11], [8, 10], [9]], Yes, 36, 36, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 17], [18, 20, 22, 24, 26, 28, 30, 32, 34], [19, 21, 23, 25, 27, 29, 31, 33, 35], [2, 16], [3, 15], [4, 14], [5, 13], [6, 12], [7, 11], [8, 10], [9]], Yes, 36, 36, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
67. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 18], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [2, 17], [3, 16], [4, 15], [5, 14], [6, 13], [7, 12], [8, 11], [9, 10]], Yes, 38, 38, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 18], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [2, 17], [3, 16], [4, 15], [5, 14], [6, 13], [7, 12], [8, 11], [9, 10]], Yes, 38, 38, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
68. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 19], [10], [2, 18], [20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [21, 23, 25, 27, 29, 31, 33, 35, 37, 39], [3, 17], [4, 16], [5, 15], [6, 14], [7, 13], [8, 12], [9, 11]], Yes, 40, 40, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 19], [10], [2, 18], [20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [21, 23, 25, 27, 29, 31, 33, 35, 37, 39], [3, 17], [4, 16], [5, 15], [6, 14], [7, 13], [8, 12], [9, 11]], Yes, 40, 40, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
69. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 20], [10, 11], [2, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [3, 18], [4, 17], [5, 16], [6, 15], [7, 14], [8, 13], [9, 12]], Yes, 42, 42, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 20], [10, 11], [2, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [3, 18], [4, 17], [5, 16], [6, 15], [7, 14], [8, 13], [9, 12]], Yes, 42, 42, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
70. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 21], [10, 12], [11], [2, 20], [22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], [23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43], [3, 19], [4, 18], [5, 17], [6, 16], [7, 15], [8, 14], [9, 13]], Yes, 44, 44, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 21], [10, 12], [11], [2, 20], [22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], [23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43], [3, 19], [4, 18], [5, 17], [6, 16], [7, 15], [8, 14], [9, 13]], Yes, 44, 44, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
71. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 22], [10, 13], [11, 12], [2, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [3, 20], [4, 19], [5, 18], [6, 17], [7, 16], [8, 15], [9, 14]], Yes, 46, 46, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 22], [10, 13], [11, 12], [2, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [3, 20], [4, 19], [5, 18], [6, 17], [7, 16], [8, 15], [9, 14]], Yes, 46, 46, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
72. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 23], [10, 14], [11, 13], [12], [2, 22], [24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47], [3, 21], [4, 20], [5, 19], [6, 18], [7, 17], [8, 16], [9, 15]], Yes, 48, 48, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 23], [10, 14], [11, 13], [12], [2, 22], [24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47], [3, 21], [4, 20], [5, 19], [6, 18], [7, 17], [8, 16], [9, 15]], Yes, 48, 48, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
73. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 24], [10, 15], [11, 14], [12, 13], [2, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [3, 22], [4, 21], [5, 20], [6, 19], [7, 18], [8, 17], [9, 16]], Yes, 50, 50, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 24], [10, 15], [11, 14], [12, 13], [2, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [3, 22], [4, 21], [5, 20], [6, 19], [7, 18], [8, 17], [9, 16]], Yes, 50, 50, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
74. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 25], [10, 16], [11, 15], [12, 14], [13], [2, 24], [26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51], [3, 23], [4, 22], [5, 21], [6, 20], [7, 19], [8, 18], [9, 17]], Yes, 52, 52, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 25], [10, 16], [11, 15], [12, 14], [13], [2, 24], [26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51], [3, 23], [4, 22], [5, 21], [6, 20], [7, 19], [8, 18], [9, 17]], Yes, 52, 52, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
75. All Conjugacy Classes Form a Complete Partition of the Group
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25], [52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26], [53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Give (classes, partition_certificate, covered_element_count, group_order, certificate).
View answer and explanation
([[0], [1, 26], [10, 17], [11, 16], [12, 15], [13, 14], [2, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [3, 24], [4, 23], [5, 22], [6, 21], [7, 20], [8, 19], [9, 18]], Yes, 54, 54, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G)
Enumerate every distinct class in source canonical order, check disjoint complete coverage and retain covered count/group order. The finite enumeration does not introduce the class equation or a centralizer count theorem. The exact result, retaining the requested order, is ([[0], [1, 26], [10, 17], [11, 16], [12, 15], [13, 14], [2, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [3, 24], [4, 23], [5, 22], [6, 21], [7, 20], [8, 19], [9, 18]], Yes, 54, 54, all_conjugacy_classes_are_equal_or_disjoint_and_cover_G).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-partition-audit · candidate
76. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
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. Element=0, other=3. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 3, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 3, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
77. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 5, 6, 7, 4], [2, 3, 0, 1, 6, 7, 4, 5], [3, 0, 1, 2, 7, 4, 5, 6], [4, 7, 6, 5, 0, 3, 2, 1], [5, 4, 7, 6, 1, 0, 3, 2], [6, 5, 4, 7, 2, 1, 0, 3], [7, 6, 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. Element=0, other=4. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 4, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 4, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
78. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 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. Element=0, other=5. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 5, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 5, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
79. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 0, 7, 8, 9, 10, 11, 6], [2, 3, 4, 5, 0, 1, 8, 9, 10, 11, 6, 7], [3, 4, 5, 0, 1, 2, 9, 10, 11, 6, 7, 8], [4, 5, 0, 1, 2, 3, 10, 11, 6, 7, 8, 9], [5, 0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10], [6, 11, 10, 9, 8, 7, 0, 5, 4, 3, 2, 1], [7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2], [8, 7, 6, 11, 10, 9, 2, 1, 0, 5, 4, 3], [9, 8, 7, 6, 11, 10, 3, 2, 1, 0, 5, 4], [10, 9, 8, 7, 6, 11, 4, 3, 2, 1, 0, 5], [11, 10, 9, 8, 7, 6, 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. Element=0, other=6. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 6, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 6, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
80. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 13, 7], [2, 3, 4, 5, 6, 0, 1, 9, 10, 11, 12, 13, 7, 8], [3, 4, 5, 6, 0, 1, 2, 10, 11, 12, 13, 7, 8, 9], [4, 5, 6, 0, 1, 2, 3, 11, 12, 13, 7, 8, 9, 10], [5, 6, 0, 1, 2, 3, 4, 12, 13, 7, 8, 9, 10, 11], [6, 0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12], [7, 13, 12, 11, 10, 9, 8, 0, 6, 5, 4, 3, 2, 1], [8, 7, 13, 12, 11, 10, 9, 1, 0, 6, 5, 4, 3, 2], [9, 8, 7, 13, 12, 11, 10, 2, 1, 0, 6, 5, 4, 3], [10, 9, 8, 7, 13, 12, 11, 3, 2, 1, 0, 6, 5, 4], [11, 10, 9, 8, 7, 13, 12, 4, 3, 2, 1, 0, 6, 5], [12, 11, 10, 9, 8, 7, 13, 5, 4, 3, 2, 1, 0, 6], [13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=7. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 7, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 7, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
81. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=8. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 8, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 8, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
82. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 11, 12, 13, 14, 15, 16, 17, 9], [2, 3, 4, 5, 6, 7, 8, 0, 1, 11, 12, 13, 14, 15, 16, 17, 9, 10], [3, 4, 5, 6, 7, 8, 0, 1, 2, 12, 13, 14, 15, 16, 17, 9, 10, 11], [4, 5, 6, 7, 8, 0, 1, 2, 3, 13, 14, 15, 16, 17, 9, 10, 11, 12], [5, 6, 7, 8, 0, 1, 2, 3, 4, 14, 15, 16, 17, 9, 10, 11, 12, 13], [6, 7, 8, 0, 1, 2, 3, 4, 5, 15, 16, 17, 9, 10, 11, 12, 13, 14], [7, 8, 0, 1, 2, 3, 4, 5, 6, 16, 17, 9, 10, 11, 12, 13, 14, 15], [8, 0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16], [9, 17, 16, 15, 14, 13, 12, 11, 10, 0, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 17, 16, 15, 14, 13, 12, 11, 1, 0, 8, 7, 6, 5, 4, 3, 2], [11, 10, 9, 17, 16, 15, 14, 13, 12, 2, 1, 0, 8, 7, 6, 5, 4, 3], [12, 11, 10, 9, 17, 16, 15, 14, 13, 3, 2, 1, 0, 8, 7, 6, 5, 4], [13, 12, 11, 10, 9, 17, 16, 15, 14, 4, 3, 2, 1, 0, 8, 7, 6, 5], [14, 13, 12, 11, 10, 9, 17, 16, 15, 5, 4, 3, 2, 1, 0, 8, 7, 6], [15, 14, 13, 12, 11, 10, 9, 17, 16, 6, 5, 4, 3, 2, 1, 0, 8, 7], [16, 15, 14, 13, 12, 11, 10, 9, 17, 7, 6, 5, 4, 3, 2, 1, 0, 8], [17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=9. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 9, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 9, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
83. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=10. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 10, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 10, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
84. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=11. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 11, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 11, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
85. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 21, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 15, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3], [15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4], [16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5], [17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6], [18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 19, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7], [19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 20, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8], [20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 21, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 22, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 23, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=12. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 12, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 12, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
86. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 19, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 20, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4], [17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5], [18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6], [19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7], [20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8], [21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9], [22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 25, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=13. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 13, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 13, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
87. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 22, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 27, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 19, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5], [19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 20, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6], [20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 21, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7], [21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 22, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8], [22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 23, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9], [23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 24, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 25, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 26, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 27, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=14. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 14, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 14, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
88. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 21, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 23, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 27, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 21, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6], [21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 22, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7], [22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 23, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8], [23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 24, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9], [24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 25, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 26, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 27, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 28, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 29, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=15. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 15, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 15, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
89. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 23, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7], [23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8], [24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9], [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 31, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=16. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 16, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 16, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
90. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 27, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 28, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 31, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 32, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 33, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 25, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8], [25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 26, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9], [26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 27, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 28, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 29, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 30, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 31, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 32, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 33, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=17. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 17, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 17, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
91. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 28, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 34, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 35, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 27, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9], [27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 28, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 29, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 30, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 31, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 32, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 33, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 34, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 35, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=18. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 18, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 18, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
92. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 31, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 34, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 35, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 36, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 37, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 29, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 30, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 31, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 32, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 33, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 34, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 35, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 36, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 37, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=19. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 19, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 19, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
93. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 31, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 35, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 36, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 37, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 38, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 39, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12, 11], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13, 12], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14, 13], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15, 14], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16, 15], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17, 16], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18, 17], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19, 18], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 19], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=20. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 20, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 20, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
94. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 35, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 37, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 38, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 39, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 40, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 41, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 33, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 34, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 35, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 36, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 37, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 38, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 39, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 40, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 41, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=21. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 21, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 21, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
95. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 35, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 36, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 37, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 38, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 39, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 40, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 41, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 42, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 43, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 35, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14, 13], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 36, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15, 14], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 37, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16, 15], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 38, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17, 16], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 39, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18, 17], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 40, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19, 18], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 41, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20, 19], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 42, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21, 20], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 43, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 21], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=22. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 22, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 22, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
96. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 38, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 40, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 41, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 42, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 43, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 44, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 45, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 37, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15, 14], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 38, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16, 15], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 39, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17, 16], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 40, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18, 17], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 41, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19, 18], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 42, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20, 19], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 43, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21, 20], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 44, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22, 21], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 45, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 22], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=23. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 23, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 23, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
97. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 47, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 47, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=24. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 24, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 24, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
98. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 42, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 43, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 44, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 45, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 46, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 47, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 48, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 49, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 41, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17, 16], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 42, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18, 17], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 43, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19, 18], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 44, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20, 19], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 45, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21, 20], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 46, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22, 21], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 47, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23, 22], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 48, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24, 23], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 49, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 24], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=25. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 25, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 25, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
99. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 43, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 44, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 45, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 46, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 48, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 49, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 50, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 51, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13], [39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14], [40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15], [41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16], [42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 43, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18, 17], [43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 44, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19, 18], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 45, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20, 19], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 46, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21, 20], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 47, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22, 21], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 48, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23, 22], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 49, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24, 23], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 50, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25, 24], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 51, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 25], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=26. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 26, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 26, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
100. A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate
Use the complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 46, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 47, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 48, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 49, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 50, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 51, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 52, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 53, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], [27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2], [29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3], [30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4], [31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5], [32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6], [33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7], [34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8], [35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9], [36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10], [37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11], [38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 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40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17], [44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 45, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18], [45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 46, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19], [46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 47, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20], [47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 48, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21], [48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 49, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22], [49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 50, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23], [50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 51, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24], [51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 52, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25], [52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 53, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 26], [53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 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. Element=0, other=27. Give (element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary).
View answer and explanation
(0, [0], Yes, 27, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)
Require a singleton class and an actual other element outside it. Check all-g commutation, then report No conjugacy to that other and its complete nonconjugacy certificate. This does not call every class a singleton or use a class-size formula. The exact result, retaining the requested order, is (0, [0], Yes, 27, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula).
Reviewed step-by-step derivation
Conjugacy class in a finite group · qf-mod45-conjugacy-singleton-nonconjugate-boundary · candidate
Canonical question JSONL · English JSONL · Validation report