Number of orbits under automorphism group

Definition

Let $G$ be a group. The number of orbits under automorphism group or number of automorphism classes of $G$ is the number of orbits of the set $G$ under the action of the automorphism group $\operatorname{Aut}(G)$.

Related notions

• Number of conjugacy classes: The conjugacy classes are the orbits under the action of the inner automorphism group, a subgroup of the automorphism group. Each conjugacy class is contained in an orbit under the automorphism group, and the number of conjugacy classes is greater than or equal to the number of orbits under automorphism group.

Facts

• Number of orbits of irreducible representations equals number of orbits under automorphism group (true for a finite group and a splitting field thereof)