Simple classical 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 cd(G) be the set of all irreducible complex character degrees of G forgetting multiplicities, that is, cd(G)={χ(1):χ∈Irr(G)} and let X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be a finite nonabelian simple classical group. In this paper, we will show that if G is a finite group and X1(G)=X1(H) then G is isomorphic to H. In particular, this implies that the nonabelian simple classical groups of Lie type are uniquely determined by the structure of their complex group algebras.