# Degree of irreducible representation divides order of group

*This article gives the statement, and possibly proof, of a constraint on numerical invariants that can be associated with a finite group*

## Statement

Let be a finite group and an irreducible representation of over an algebraically closed field of characteristic zero. Then, the degree of divides the order of .

## Proof

### Introduction of some algebraic integers

`Further information: Convolution algebra on conjugacy classes`

Using the convolution algebra on conjugacy classes, we can show that for any representation with character , and any conjugacy class , the number:

are algebraic integers. Note that is the degree of .

### A little formula

We know that if is irreducible:

by the ortho*normality* of the irreducible characters.

Dividing by , we get:

Note that since the characters are algebraic integers and so are the values taken by , the overall left-hand side is an algebraic integer. Thus is an algebraic integer.

But since it is a rational number, it must be a rational integer, or in other words, the degree of divides the order of .