On the conjugacy class exponent of the finite simple groups

TL;DR

This paper establishes upper bounds on the conjugacy class exponent for all finite simple groups, revealing that most have an exponent at most 8, with specific larger bounds for certain Lie type groups.

Contribution

It provides the first comprehensive bounds on the conjugacy class exponent for all finite simple groups, including detailed bounds for Lie type groups.

Findings

Most finite simple groups have conjugacy class exponent at most 8.

Specific larger bounds are given for groups like PSL, PSU, and exceptional groups.

Bounds are also provided for semisimple and unipotent elements in Lie type groups.

Abstract

The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g \in G\}$. We provide upper bounds for $e(G)$ for every finite simple group $G$. In particular, we show that $e(G)\leq 8$ unless $G\in\{\mbox{PSL}_n(q), \mbox{PSU}_n(q), E_6(q),{^2}E_6(q)\}$. For the latter groups $e(G)\leq n,3n+3,36,36$, respectively. In addition, we bound from above the generalized order of semisimple and unipotent elements of finite simple groups of Lie type.