Recognition by element orders for simple linear and unitary groups

Abstract

For a finite group $G$, let $\ddot{I}‰(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $\ddot{I}‰(H)=\ddot{I}‰(G)$. 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 $\ddot{I}‰(H)=\ddot{I}‰(G)$ are described. We complete the solution of the recognition problem for the finite simple linear and unitary groups.