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

From Groupprops

## Definition

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

- The number of orbits under automorphism group of the elements of , or equivalently, the number of orbits of the conjugacy classes of under the action of the automorphism group.
- 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

## Facts used

## 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.