For $G$ a group and $g \in G$ an element, the conjugacy class of $g$ is the orbit of $g$ under the adjoint action of $G$ on itself, hence the subset $\{ h g h^{-1} | h \in G \} \subset G$ of all elements in $G$ obtained from $g$ by conjugation with another group element.

Examples

The conjugacy class of the neutral element consists of precisely the neutral element itself.

In an abelian group, conjugacy classes are singletons, one for each element of the group.

Properties

For discrete groups

For any finite group the number of its conjugacy classes is equal to the number of its irreducible representations. For finite groups of Lie type this result can be strengthened to show that, in heuristic terms, there is a canonical way to match conjugacy classes of a group $G$ to the irreducible representations of its dual group $G^{\ast}$. The dual group is defined in terms of the root datum? of $G$ in Deligne-Lusztig theory.