For a group and an element, the conjugacy class of is the orbit of under the adjoint action of on itself, hence the subset of all elements in obtained from by conjugation with another group element.
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.
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 to the irreducible representations of its dual group . The dual group is defined in terms of the root datum? of in Deligne-Lusztig theory.