The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple groups by spectrum.
Dedicated to 70-th anniversary of V. D. Mazurov Review: 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 = 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.
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