Let $\chi$ be the character of a representation of a simple group $G$ and let $g\in G$. If $g$ has order two and $G\neq C_2$ then show that $\chi(g)\equiv \chi (e)$ modulo 4. The hint I get is to consider the action of the determinant of $g$ acting on $V$.
$g$ has order two so the eigenvalues of $g$ must all be $\pm 1$. So I think somehow I need to show $\det (g)=1$. But how?