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

Abstract

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let $${{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}}$$ be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X1(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 $${{\rm cd}(S)\subseteq {\rm cd}(H)}$$ then S must be isomorphic to H. As a consequence, we show that if G is a finite group with $${{\rm X}_1(G)\subseteq {\rm X}_1(H)}$$ 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.