Extending Huppert’s conjecture to almost simple groups of Lie type

Abstract

Let $G$ be a finite group and $\mathrm{cd}\left(G\right)$ be the set of all irreducible complex character degrees of $G$ without multiplicities. The aim of this paper is to propose an extension of Huppert’s conjecture from non-Abelian simple groups to almost simple groups of Lie type. Indeed, we conjecture that if $H$ is an almost simple group of Lie type with $\mathrm{cd}\left(G\right)=\mathrm{cd}\left(H\right)$, then there exists an Abelian normal subgroup $A$ of $G$ such that $G/A\cong H$. It is furthermore shown that $G$ is not necessarily the direct product of $H$ and $A$. In view of Huppert’s conjecture, we also show that the converse implication does not necessarily hold for almost simple groups. Finally, in support of this conjecture, we will confirm it for projective general linear and unitary groups of dimension 3.