Martin Isaacs's exercise 3.6 (character theory of finite groups)

Question

I'm trying to solve this exercise, can anyone help me?

Let $G$ be a p-group, and suppose $\chi\in{Irr(G)}$. Show that $\chi(1)^2$ divides $|G:Z(\chi)|$

Thanks a lot.

Answer 1

Hint: Use Corollary 2.30 and theorem 3.12.

By Corollary 2.30, $\chi(1)^2 \leq [G:Z(\chi)]$. By Theorem 3.12, $\chi(1)$ divides $[G:Z(\chi)]$ so is a power of a prime. However $\leq$ is the same as divides for two powers of the same prime. $\square$