Finite simple groups have many classes of $p$-elements

TL;DR

This paper establishes an upper bound on the order of finite nonabelian simple groups based on the maximum number of automorphism classes of p-elements, extending previous results and impacting the study of global field extensions.

Contribution

It introduces a bound on the size of simple groups in terms of automorphism classes of p-elements, generalizing earlier work and linking to number theory.

Findings

Bound on |T| in terms of m(T)

Extension of previous results by Pyber and Héthelyi-Külshammer

Implications for relative Brauer groups of global fields

Abstract

For an element $x$ of a finite group $T$, the $\mathrm{Aut}(T)$-class of $x$ is the set $\{ x^\sigma\mid \sigma\in \mathrm{Aut}(T)\}$. We prove that the order $|T|$ of a finite nonabelian simple group $T$ is bounded above by a function of the parameter $m(T)$, where $m(T)$ is the maximum, over all primes $p$, of the number of $\mathrm{Aut}(T)$-classes of elements of $T$ of $p$-power order. This bound is a substantial generalisation of results of Pyber, and of H\'ethelyi and K\"ulshammer, and it has implications for relative Brauer groups of finite extensions of global fields.