A counterexample to a conjecture of A. R. Miller

Abstract

Let $χ$ be an irreducible character of a finite group $G$. A. R. Miller conjectured that the proportion of elements $g\in G$ such that $χ(g)$ is zero or a root of unity is at least 1/2. We construct a character of a perfect group of order 69120 such that this proportion is 511/1152.