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.