Degree of irreducible representation divides order of group
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 .
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 orthonormality 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 .