Let V be an n-dimensional irriducible complex representation of a finite group G, let C be it's center. Show that $|\chi (s)| = n $ when $s \in C$. Where $\chi$ is the character function.
My attempt at proof:
Given that $\rho_s \rho_g = \rho_g\rho_s$ when $g \in G, s \in C$, by Schur's Lemma we have that $\rho_s=\lambda \ Id$, $\lambda \in \mathbb{C}$. Therefore $\chi(s)=Tr(\rho_s)=\lambda n$
My question is why does $|\lambda|=1$?
I have been unable to prove this any further so any hints or help would be greatly appreciated.
Since $s$ is an element of the finite group $G$, we have $s^k=1$ for some positive integer $k$. This means that for your $\lambda$ we must have $\lambda^k=1$, whence $\lambda$ is a root of unity, so $|\lambda|=1$.