Vanishing Elements of Prime Power Order

Abstract

An element $x$ in a finite group $G$ is said to be \textit{vanishing} if some (complex) irreducible character of $G$ takes value $0$ at $x$. In this article, we prove that every non-abelian finite simple group, except $\mathrm{SL}_2(4)$ and $\mathrm{SL}_2(8)$, contains a vanishing element \textit{of prime power order} whose conjugacy class size is divisible by three distinct primes. We use this result to obtain the following generalization of a result of Robati ($2021$): If $G$ is a non-solvable finite group in which, the conjugacy class size of all the vanishing elements of prime power order has at most two distinct prime divisors, then $G/\mathrm{Sol}(G)$ is a direct product of mutually isomorphic simple groups among $\mathrm{SL}_2(4)$ and $\mathrm{SL}_2(8)$. ($\mathrm{Sol}(G)$ is the largest normal solvable subgroup of $G$.)