Huppert’s conjecture and almost simple groups

Let G be a finite group and cd(G) denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group whose socle is H0 = PSL(2, q) with q = 2 f (f prime) such that cd(G) = cd(H), then there exists an abelian subgroup A of G such that G/A is isomorphic to H. In view of Huppert’s conjecture (2000), the main result of this paper gives rise to some examples that G is not necessarily a direct product of A and H, and consequently, we cannot extend this conjecture to almost simple groups. Mathematics Subject Classification (2010). Primary: 20C15; Secondary: 20D05