1
$\begingroup$

Let $G$ be a finite $p$-group. Is it true that every irreducible representation of $G$ over an algebraically closed field of characteristic zero ($\mathbb{C}$, for example) must have dimension a power of $p$?

Proof or reference are appreciated.

1 Answers 1

5

The degrees of the simple complex representations of a group divide the order of the group, so yes. (In fact, they divide the index of the center of the group, which in your case is smaller.)

See, for example, Serre's book for a proof.

  • 0
    Do you know what page/what chapter of Serre's book it is in?2016-12-15