Finite simple groups have many classes of prime order elements

TL;DR

This paper improves a known bound on finite simple groups by demonstrating that the maximum number of conjugacy classes of prime order elements determines the group's order, replacing prime-power with prime order in the bound.

Contribution

It strengthens previous results by showing that the bound depends on classes of prime order elements, not just prime-power order elements.

Findings

The order of a finite simple group is bounded by the number of conjugacy classes of prime order elements.

Prime order classes suffice to determine the group's size, improving earlier bounds.

The result refines understanding of the structure of finite simple groups.

Abstract

Let $T$ be a finite non-abelian simple group. Giudici, Morgan and Praeger have shown that the order of $T$ is bounded above by a function depending on the maximum number of $\mathrm{Aut}(T)$-classes of elements of $T$ of prime-power order. In this note, we strengthen this result by showing, in particular, that prime-power can be replaced by prime.