TITLE: Eigenvalues of representations
QUESTION [5 upvotes]: Let $\rho$ be a representation of $G$ on $V$.  Why are its eigenvalues roots of unity?

REPLY [10 votes]: I assume $G$ is finite.  In that case any $g \in G$ has some finite order $n$, hence $\rho(g)^n = 1$.  It follows that the characteristic polynomial of $\rho(g)$ divides $x^n - 1$.