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

Michael Giudici, Luke Morgan and Cheryl E. Praeger

Vol. 336 (2025), No. 1-2, 137–160

DOI: 10.2140/pjm.2025.336.137

Abstract

For an element $x$ of a finite group $T$, the $\mathrm{Aut}<!--FUNCTION\; APPLICATION-->(T)$-class of $x$ is $\{x^{\sigma }\mid \sigma \in \mathrm{Aut}<!--FUNCTION\; APPLICATION-->(T)\}$. We prove that the order $\left|T\right|$ 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 $\mathrm{Aut}<!--FUNCTION\; APPLICATION-->(T)$-classes of elements of $T$ of $p$-power order. This bound is a substantial generalisation of the results of Pyber  (1992) and of Héthelyi and Külshammer (2005), and it has implications for relative Brauer groups of finite extensions of global fields.

Dedicated with admiration and thanks to the memory of our colleague Gary M. Seitz

Keywords

finite simple group, $p$-elements, conjugacy classes, order bounds

Mathematical Subject Classification

Primary: 20D06, 20E32

Milestones

Received: 31 July 2024

Revised: 26 November 2024

Accepted: 27 November 2024

Published: 26 May 2025