Huppert’s conjecture and almost simple groups

Abstract

Let G be a finite group and \mathsf{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 H_{0}=\mathrm{PSL}(2,q) with q=2^{f} ( f prime) such that \mathsf{cd}(G) =\mathsf{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.