The problem of recognition of finite simple groups by element orders is solved

Abstract

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