Abstract
Given a finite group G, denote by N(G) the set of sizes of its conjugacy classes. We show that if G has trivial center and N(G) equals or
for
then G has a composition factor isomorphic to the alternating group Altk for some
such that the half-interval
contains no primes. As a corollary, we prove Thompson’s conjecture is valid for simple alternating groups.
2010 MATHEMATICS SUBJECT CLASSIFICATION: