Vanishing Elements of Prime Power Order

TL;DR

This paper investigates vanishing elements of prime power order in finite groups, proving their existence in most non-abelian simple groups and exploring implications for the structure of non-solvable groups with restricted conjugacy class sizes.

Contribution

It establishes the existence of vanishing elements of prime power order with specific properties in most non-abelian simple groups and generalizes prior results on the structure of certain non-solvable groups.

Findings

Most non-abelian simple groups contain vanishing elements of prime power order with conjugacy class size divisible by three primes.

A classification of non-solvable groups where vanishing elements have conjugacy class sizes with at most two prime divisors.

Identification of specific simple groups ($ ext{SL}_2(4)$ and $ ext{SL}_2(8)$) related to the structure of certain non-solvable groups.

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 $G$.)