Recognition by element orders for simple linear and unitary groups

TL;DR

This paper completes the recognition problem for finite simple linear and unitary groups by analyzing their element order sets and classifying groups with identical element order spectra.

Contribution

It provides a complete solution to the recognition problem for these groups, determining the number and structure of groups sharing the same element order set.

Findings

The recognition problem is solved for finite simple linear groups.

The recognition problem is solved for finite simple unitary groups.

All groups with the same element order set as these groups are classified.

Abstract

For a finite group $G$, let $\omega(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $\omega(H)=\omega(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if $h(G)$ is finite, then all finite groups $H$ with $\omega(H)=\omega(G)$ are described. We complete the solution of the recognition problem for the finite simple linear and unitary groups.