# Degree of irreducible representation over field of characteristic coprime to order divides product of order and Euler totient function of exponent

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## Contents

## Statement

Suppose is a finite group, is a field whose characteristic is relatively prime to the order of , and is an irreducible representation of over (which need not be absolutely irreducible, i.e., it may split over some algebraic extension of ). Then, the degree of divides the product of the order of and the Euler totient function of the exponent of .

## Related facts

### Other facts in general

- Degree of irreducible representation of nontrivial finite group is strictly less than order of group
- Maximum degree of irreducible real representation is at most twice maximum degree of irreducible complex representation

### Facts over a splitting field

We can obtain much better bounds on the degrees of irreducible representations if is a splitting field for . Specifically:

- Degree of irreducible representation divides group order
- Degree of irreducible representation divides order of inner automorphism group
- Degree of irreducible representation divides index of abelian normal subgroup
- Order of inner automorphism group bounds square of degree of irreducible representation
- Sum of squares of degrees of irreducible representations equals order of group
- Number of irreducible representations equals number of conjugacy classes

## Facts used

- Sufficiently large implies splitting: Any sufficiently large field for a finite group , i.e., a field that contains all the roots of unity where is the exponent of , is also a splitting field for .
- Degree of irreducible representation divides group order