  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 $n\neq 6, 10$. 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
