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 $q=2^{2m+1}geq 8$‎, ‎then $G$ is isomorphic to $Sz(q)$‎. ‎This addresses to the well-known J‎. ‎G‎. ‎Thompson's problem (1987) for simple groups‎.