Zeros of irreducible characters of metabelian $p$-groups

Abstract

We show that if $\chi$ is an irreducible complex character of a metabelian $p$-group $P$, where $p$ is an odd prime, and if $x\in P$ satisfies $\chi(x)\neq 0$, then the order of $x$ divides $|P|\chi(1)^2.