On Characterization of a Finite Group with Non-simple Socle by the Set of Conjugacy Class Sizes

Let G be a finite group and N(G) be the set of its conjugacy class sizes without 1. In the 1980s Thompson conjectured that the equality N(G) = N(S), where Z(G) = 1 and S is simple, implies the isomorphism G S. In a series of papers of different authors, Thompson’s conjecture was proved. We show that if G is a finite centerless group with the same set of conjugacy class sizes as L2(q) × L2(q), where q is odd greater 3 and numbers q − 1, q + 1 are not power of 2, then G L2(q) × L2(q).