Finite groups of the same type as Suzuki groups

Abstract

For a finite group $G$ and a positive integer $n$, let $G(n)$ be the set of all elements in $G$ such that $x^n=1$. The groups $G$ and $H$ are said to be of the same (order) type if $G(n)=H(n)$, for all $n$. The main aim of this paper is to show that if $G$ is a finite group of the same type as Suzuki groups $Sz(q)$, where $q=2^{2m+1}\ge 8$, then $G$ is isomorphic to $Sz(q)$. This addresses the well-known J. G. Thompson's problem (1987) for simple groups.