On characterization of a finite group by the set of conjugacy class sizes

Abstract

Let [Formula: see text] be a finite group and [Formula: see text] be the set of its conjugacy class sizes. In the 1980s, Thompson conjectured that the equality [Formula: see text], where [Formula: see text] and [Formula: see text] is simple, implies the isomorphism [Formula: see text]. In a series of papers of different authors, Thompson’s conjecture was proved. In this paper, we show that in some cases it is possible to omit the conditions [Formula: see text] and [Formula: see text] is simple and prove a more general result.