Finite groups of the same type as Suzuki groups

TL;DR

This paper proves that any finite group sharing the same element order structure as a Suzuki group is actually isomorphic to it, solving a longstanding problem in the classification of finite simple groups.

Contribution

It establishes that finite groups of the same type as Suzuki groups are necessarily isomorphic to them, confirming a conjecture related to Thompson's problem.

Findings

Finite groups of the same type as Suzuki groups are isomorphic to them.

Addresses Thompson's problem for simple groups.

Provides a characterization of Suzuki groups based on element orders.

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 the well-known J. G. Thompson's problem (1987) for simple groups.