On groups with the same character degrees as almost simple groups with socle Mathieu groups

Abstract

Let $G$ be a finite group and $\mathrm{cd}(G)$ 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 a Mathieu group such that $\mathrm{cd}(G)=\mathrm{cd}(H)$, then there exists an abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. In view of Huppert's conjecture (2000), we also provide some examples to show that $G$ is not necessarily a direct product of $A$ and $H$, and hence we cannot extend this conjecture to almost simple groups.