Recognizability of alternating groups by spectrum

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 $G$. It is proved that simple alternating groups An are recognizable by spectrum, for $n\ne6,10$. This implies that every finite group whose spectrum coincides with that of a finite non-Abelian simple group has at most one non-Abelian composition factor.