The set of elements in a group obtained by conjugating a fixed element by every group element; formally, for g in G the conjugacy class is {aga^{-1} : a in G}.

Principle

Conjugacy groups elements that share the same internal position relative to the group structure; members are equivalent under the group's inner automorphisms.

Demonstration

In the symmetric group S3, the 3‑cycles form a conjugacy class: { (1 2 3), (1 3 2) } because conjugation by any permutation permutes cycle notation to another 3‑cycle.

Misapplication

Assuming every conjugacy class is a subgroup — in general a conjugacy class is not closed under the group operation and therefore not a subgroup except in special cases (e.g., class of the identity).

Consequence

Conjugacy classes partition the group and are central to representation theory and class equations; elements in the same class have the same order and trace in any representation.

Reversal

A normal subgroup is a union of conjugacy classes; inverting the perspective, invariance under conjugation (normality) organizes classes into subgroup structure.

Boundary

Defined only within a group context and for inner conjugation by group elements; does not apply to arbitrary sets with an external action unless that action arises from conjugation in a group.

Semantic Tension

Often contrasted with cosets: cosets are translates by a subgroup and need not partition by element equivalence under inner automorphisms, while conjugacy classes partition by conjugation equivalence.

Synthesis

A conjugacy class is the equivalence class of an element under inner automorphisms, partitioning a group into sets of elements that are the same up to relabeling by group symmetries.