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

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

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

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

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

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

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

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

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.