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

From Groupprops

## Statement

Suppose is a finite group, is a splitting field for , and is a subgroup of the automorphism group of . Denote by the set of conjugacy classes in and by the set of (equivalence classes of) irreducible representations of over . Then, acts naturally on both and .

The claim is that the number of orbits under the action of on equals the number of orbits under the action of on .

## Related facts

### Similar facts

- Number of irreducible representations equals number of conjugacy classes (extreme case where the subgroup has no outer automorphisms)
- Number of orbits of irreducible representations equals number of orbits under automorphism group (extreme case where the subgroup is the whole automorphism group).
- 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

- Application of Brauer's permutation lemma to group automorphism on conjugacy classes and irreducible representations, which in turn uses Brauer's permutation lemma (actually, we don't need the full strength of the statement about cycle types for our purpose, we simply need the
*fixed point*version, which can be deduced even more directly) - Orbit-counting theorem (also called Burnside's lemma)

## Proof

**Given**: A finite group , a splitting field for , a subgroup of the automorphism group of . denotes the set of conjugacy classes of , denotes the set of equivalence classes of irreducible representations of over .

**To prove**: The number of orbits of under the natural action of equals the number of orbits of under the action of .

**Proof**:

No. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | For any , the cycle type for the permutation induced by on is the same as the cycle type of the permutation induced by on . | Fact (1) | Fact-direct | ||

2 | For any , the number of fixed points for the permutation induced by on equals the number of fixed points of the permutation induced by on . | Step (1) | Follows from the previous step, since the fixed points are just the length one cycle. | ||

3 | The number of orbits in under the action of equals the number of orbits in under the action of . | Fact (2) (orbit-counting theorem) | Step (2) | [SHOW MORE] |