# if $G$ is a finite group, can we put a bound on the dimension of its irreducible representations?

Suppose the order of $G$ is $n < \infty$. Suppose $R$ is an irreducible representation of $G$, what can we conclude about its dimension with relation to the order of $G$?

• The sum of the squares of the dimensions of the irreducible representations of $G$ over $\Bbb C$ is just $\#G$, so, crudely, any representation has dimension $\leq \sqrt{\#G}$. It's also true that the dimension of any such representation divides $\#G$. (Note that these things need not be true for representations over nonclosed fields, e.g., $\Bbb Q$.) Apr 18 '17 at 16:19