Simple exceptional groups of Lie type are determined by their character degrees

Abstract

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$ forgetting multiplicities, and let $\textrm{X}_1(G)$ be the set of all irreducible complex character degrees of $G$ counting multiplicities. Let $H$ be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if $S$ is a non-abelian simple group and $\textrm{cd}(S)\subseteq \textrm{cd}(H)$ then $S$ must be isomorphic to $H.$ As a consequence, we show that if $G$ is a finite group with $\textrm{X}_1(G)\subseteq \textrm{X}_1(H)$ then $G$ is isomorphic to $H.$ In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.