# Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have $$d_{\alpha}|\#G,$$ where $d_\alpha$ is the degree of the representation and $\#G$ is the order of the group.

I learnt one proof from Barry Simon's Representation of Finite and Compact Groups. However his proof depends on some delicate arguments in algebraic number theory, which seems to be very surprising and not so illuminating.

I am wondering whether this is necessary because $d_{\alpha}|\#G$ seems naive enough to be proved entirely in the framework of representation theory.

I am wondering whether someone can share a hint. If there is no such proof yet, can we at least try to understand, within representation theory, why this should be true?

Thanks very much!

• This question is very similar to math.stackexchange.com/questions/243221 – Derek Holt Apr 27 '13 at 18:07
• @DerekHolt Nice link! But it seems answers there are not really satisfactory for they also heavily use algebraic number theory. – Hui Yu Apr 28 '13 at 15:05