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 A are recognizable by spectrum, for n ≠ 6, 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. [ABSTRACT FROM AUTHOR]