Number of orbits of irreducible representations equals number of orbits under automorphism group

Definition

Suppose $G$ is a group and $K$ is a splitting field for $G$. Then, the following two numbers are equals:

1. The number of orbits under automorphism group of the elements of $G$, or equivalently, the number of orbits of the conjugacy classes of $G$ under the action of the automorphism group.
2. The number of orbits of irreducible representations under the action of the automorphism group.

Related facts

Similar facts

• Number of irreducible representations equals number of conjugacy classes
• Number of orbits of irreducible representations equals number of orbits of conjugacy classes under any subgroup of automorphism group
• Application of Brauer's permutation lemma to group automorphism on conjugacy classes and irreducible representations
• Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Opposite facts

• Orbit sizes for irreducible representations may differ from orbit sizes for conjugacy classes under action of automorphism group

Facts used

1. Number of orbits of irreducible representations equals number of orbits of conjugacy classes under any subgroup of automorphism group

Proof

The proof follows directly from Fact (1), and the observation that the number of orbits of conjugacy classes equals the number of orbits of elements when we take the whole automorphism group, because each orbit of elements under the whole automorphism group is a union of conjugacy classes.