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

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G) = {χ(1) {pipe} χ ∈ Irr(G)} be the set of all irreducible complex character degrees of G forgetting multiplicities, and let 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 cd(S) ⊆ cd(H) then S must be isomorphic to H. As a consequence, we show that if G is a finite group with X ₁ (G) ⊆ 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.