THE NUMBER OF DISTINCT EIGENVALUES OF ELEMENTS IN FINITE LINEAR GROUPS

Abstract

Let $G$ be a finite (non-abelian) irreducible linear subgroup over the complex numbers, and let $g$ be an element of $G$ of prime order $p$. Suppose that $g$ does not belong to a proper normal subgroup of $G$. We show that the number of distinct eigenvalues of $g$ can only be $p,p-1,p-2,(p+1)/2$ or $(p-1)/2$. Moreover, we provide a full classification of such groups $G $ provided that $g$ has at most $p-2$ distinct eigenvalues.