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Question 24887: A Singleton Class Commutes with Every Group Element, while an Outside Element Is Nonconjugate

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

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← Question derivation index · Original question batch

R0048-24887 · q-20261015-advance-20261114-r03-finite-group-conjugacy-class-0087

Original question and givens

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).

  • G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [conjugacy-singleton-nonconjugate-boundary]

After

derived = [[element, singleton_class, commutes_with_all, other, conjugate_to_other, nonconjugate_certificate, boundary]]

Conditions: none

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

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [Conjugacy ORIGINAL group, 64, [[0, 1, 2, 3, 4, 5, 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]], 0]

After

derived = [28, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21], [22, 22, 22, 22], [23, 23, 23, 23], [24, 24, 24, 24], [25, 25, 25, 25], [26, 26, 26, 26], [27, 27, 27, 27]], Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001

S003 · implies

Before

source roles = [Conjugacy ORIGINAL group, 28, FOR EVERY x,y,z in0..n-1]

After

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 21952, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002

S004 · implies

Before

source roles = [Conjugacy ORIGINAL group, 0]

After

derived = [[0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [[0, 0, 0, 0], [1, 13, 0, 0], [2, 12, 0, 0], [3, 11, 0, 0], [4, 10, 0, 0], [5, 9, 0, 0], [6, 8, 0, 0], [7, 7, 0, 0], [8, 6, 0, 0], [9, 5, 0, 0], [10, 4, 0, 0], [11, 3, 0, 0], [12, 2, 0, 0], [13, 1, 0, 0], [14, 14, 0, 0], [15, 15, 0, 0], [16, 16, 0, 0], [17, 17, 0, 0], [18, 18, 0, 0], [19, 19, 0, 0], [20, 20, 0, 0], [21, 21, 0, 0], [22, 22, 0, 0], [23, 23, 0, 0], [24, 24, 0, 0], [25, 25, 0, 0], [26, 26, 0, 0], [27, 27, 0, 0]], Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Group_ALL_two_sided_inverses · Uses: G001, S003

S005 · implies

Before

source roles = [0, 0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]]

After

derived = [[0, 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, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S004

S006 · implies

Before

source roles = [1, 13, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S005

S007 · implies

Before

source roles = [2, 12, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S006

S008 · implies

Before

source roles = [3, 11, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S007

S009 · implies

Before

source roles = [4, 10, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S008

S010 · implies

Before

source roles = [5, 9, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S009

S011 · implies

Before

source roles = [6, 8, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S010

S012 · implies

Before

source roles = [7, 7, [0, 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]]

After

derived = [[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], [0, 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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S011

S013 · implies

Before

source roles = [8, 6, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S012

S014 · implies

Before

source roles = [9, 5, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S013

S015 · implies

Before

source roles = [10, 4, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S014

S016 · implies

Before

source roles = [11, 3, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S015

S017 · implies

Before

source roles = [12, 2, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S016

S018 · implies

Before

source roles = [13, 1, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S017

S019 · implies

Before

source roles = [14, 14, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S018

S020 · implies

Before

source roles = [15, 15, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S019

S021 · implies

Before

source roles = [16, 16, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S020

S022 · implies

Before

source roles = [17, 17, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S021

S023 · implies

Before

source roles = [18, 18, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S022

S024 · implies

Before

source roles = [19, 19, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S023

S025 · implies

Before

source roles = [20, 20, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S024

S026 · implies

Before

source roles = [21, 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, 24, 25, 26, 27]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S025

S027 · implies

Before

source roles = [22, 22, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S026

S028 · implies

Before

source roles = [23, 23, [0, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S027

S029 · implies

Before

source roles = [24, 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, 24, 25, 26, 27]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S028

S030 · implies

Before

source roles = [25, 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]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S029

S031 · implies

Before

source roles = [26, 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, 26, 27]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S030

S032 · implies

Before

source roles = [27, 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, 27]]

After

derived = [[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], [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]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_EVERY_g_EVERY_element_exact_action_row · Uses: G001, S031

S033 · implies

Before

source roles = [0, 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, 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]]]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_ALL_action_axioms_and_GENERAL_algebra · Uses: G001, S032

S034 · implies

Before

source roles = [[[0, 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]]]

After

derived = [[[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]], 28, 28, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_complete_orbits_equal_or_disjoint_partition · Uses: G001, S033

S035 · implies

Before

source roles = [0]

After

derived = [[[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0], [16, 0], [17, 0], [18, 0], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0]], [0]]

Conditions: none

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

Step identifiers and dependencies

Rule: Conjugacy_actual_element_ALL_images_not_coset · Uses: G001, S034

S036 · implies

Before

source roles = [0, 14, [0]]

After

derived = [[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, Yes], [9, 9, 9, Yes], [10, 10, 10, Yes], [11, 11, 11, Yes], [12, 12, 12, Yes], [13, 13, 13, Yes], [14, 14, 14, Yes], [15, 15, 15, Yes], [16, 16, 16, Yes], [17, 17, 17, Yes], [18, 18, 18, Yes], [19, 19, 19, Yes], [20, 20, 20, Yes], [21, 21, 21, Yes], [22, 22, 22, Yes], [23, 23, 23, Yes], [24, 24, 24, Yes], [25, 25, 25, Yes], [26, 26, 26, Yes], [27, 27, 27, Yes]], Yes, No, Yes]

Conditions: none

Right multiplying gxg^-1=x by g yields gx=xg,so singleton is equivalent to all-g commutation.Other is actually outside the complete orbit,justifying nonconjugacy without declaring every class singleton.

Step identifiers and dependencies

Rule: Conjugacy_singleton_iff_ALL_commute_and_actual_other_outside · Uses: G001, S035

Reverse checks

  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(0, [0], Yes, 14, No, Yes, singleton_class_iff_commutes_with_every_g_and_other_outside_class_is_nonconjugate_without_class_size_formula)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 02827ae19f6c158f28a6cc5aed7bfd033e9b0330e409a71a7c2fad5198567ff1
  • candidate record SHA-256: 321413098e5458dcd9c133a5b8abd996be1d9fbcd4d05add1275c70a50c3bb22
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON