GROUPS WITH THE SAME CHARACTER DEGREES AS SPORADIC ALMOST SIMPLE GROUPS

Abstract

Let$G$be a finite group and$\mathsf{cd}(G)$denote the set of complex irreducible character degrees of$G$. We prove that if$G$is a finite group and$H$is an almost simple group whose socle is a sporadic simple group$H_{0}$and such that$\mathsf{cd}(G)=\mathsf{cd}(H)$, then$G^{\prime }\cong H_{0}$and there exists an abelian subgroup$A$of$G$such that$G/A$is isomorphic to$H$. In view of Huppert’s conjecture, we also provide some examples to show that$G$is not necessarily a direct product of$A$and$H$, so that we cannot extend the conjecture to almost simple groups.