##  [Conjugacy](/conjugacy-0) 

 Definition

A binary relation on the elements of a group (or an algebraic structure with invertible elements) in which two elements a and b are conjugate if there exists an invertible g with a = g b g^{-1}; equivalence classes under this relation are called conjugacy classes.

 

 

 

 

 

 





## Principle

Principle

Conjugacy is governed by inner automorphisms: conjugating by an element defines an isomorphism from the group to itself, and these inner maps partition the group into orbits (conjugacy classes) whose stabilizers are centralizers.

 

 

 

 

 





## Demonstration

Demonstration

In the symmetric group S3, all transpositions are conjugate: for example (12) = (13)(23)(13)^{-1}, so the three transpositions form a single conjugacy class; in GL(n) two matrices are conjugate exactly when they represent the same linear map in different bases (similarity).

 

 

 

 

## Misapplication

Misapplication

Treating conjugacy as literal equality or as commutativity — for instance claiming a and b conjugate implies a = b, or that conjugate elements commute — is incorrect and hides the group action nature of conjugation.

 

 

 

 

 





## Consequence

Consequence

Correct use of conjugacy yields invariants and classification tools: class functions, character tables, the center (elements conjugate only to themselves), and the relationship between conjugacy classes and normal subgroups via union-of-classes criteria.

 

 

 

 

## Reversal

Reversal

The inversion of the concept is centrality: instead of relating elements by a change of coordinates, one can ask when elements are fixed by all conjugations (belong to the center), or conversely study cosets that identify elements without inner conjugation.

 

 

 

 

 





## Boundary

Boundary

Conjugacy is defined in settings with a notion of invertibility (groups, groupoids, invertible linear operators); naïve analogues in semigroups or noninvertible contexts require adjusted definitions and may not produce equivalence relations with the same properties.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Conjugacy competes semantically with similarity in linear algebra and with general equivalence relations: similarity is a specific instance (matrices under change of basis), while other equivalences (e.g., orbit equivalence under different group actions) can be mistaken for conjugacy if the acting group is not inner.

 

 

 

 

 





## Synthesis

Synthesis

Conjugacy packages the idea of 'same up to change of coordinates' into an equivalence relation generated by inner automorphisms: it identifies elements that behave the same inside the group action, organizes structure into classes and centralizers, and furnishes the language for invariants used in classification and representation theory.