Conjugacy class

Definition

Let $G$ be a group and let $a \in G$. The conjugacy class of $a$ is the set $\operatorname{Cl}(a) = \{gag^{-1} : g \in G\}$.

Remarks

The conjugacy classes are the equivalence classes of the relation "is conjugate to", so they partition $G$. They are the orbits of the action by conjugation. The class of $a$ is $\{a\}$ if and only if $a$ lies in the centre of $G$. In the symmetric group $S_n$, the conjugacy classes are given by the cycle types.

Explanations

L1 human

The conjugacy class of an element collects all the elements that look like it after a renaming by conjugation. In an abelian group, each element is alone in its class. In $S_3$, the three transpositions form one class, and the two $3$-cycles form another class.