Vanishing Elements of Prime Power Order
Sonakshee Arora, Rahul Dattatraya Kitture
TL;DR
The paper investigates how vanishing elements of prime power order constrain the structure of finite groups. It proves that every non-abelian finite simple group $G$ with $G\not\cong \mathrm{SL}_2(4)$ or $\mathrm{SL}_2(8)$ contains a vanishing element of prime power order whose conjugacy class size $|x^G|$ is divisible by three distinct primes, and it extends Robati's result by showing that if, in a non-solvable group $G$, all such $|x^G|$ are divisible by at most two primes, then $G/\mathrm{Sol}(G)$ is a direct product of copies of $\mathrm{SL}_2(4)$ or $\mathrm{SL}_2(8)$. The approach combines a detailed case analysis across the Classification of Finite Simple Groups, constructing elements with large and prime-diverse conjugacy classes using maximal tori and Zsigmondy primes, and leverages Clifford theory to lift vanishing properties to larger groups. The results connect character-theoretic vanishing properties to global group structure, narrowing the possibilities for non-solvable groups under prime-divisor constraints on vanishing-class sizes and generalizing prior work in this area.
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 $G$.)