Zeros and Roots of Unity for Characters of Solvable Groups

Abstract

We prove Miller's conjectured bound for solvable groups: if $G$ is a finite solvable group and $χ\in\Irr(G)$, then $\chi (g)$ is zero or a root of unity for at least half of the elements $g\in G$. We also prove the same bound for monomial irreducible characters of arbitrary finite groups.