Recognizability by spectrum of alternating groups

Abstract

The spectrum of a group is the set of its element orders. A finite group $G$ is said to be recognizable by spectrum if every finite group that has the same spectrum as $G$ is isomorphic to $G$. We prove that the simple alternating groups $A_n$ are recognizable by spectrum when $n\ne 6,10$. This implies that every finite group with the same spectrum as that of a finite nonabelian simple group, has at most one nonabelian composition factor