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 cd(G) = {χ(1) | 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.

A new characterization for the simple group PSL(2, p 2) by order and some character degrees

“…It was conjectured that all nonabelian simple groups are uniquely determined by the structure of their complex group algebras. This conjecture was verified in [9], [12], [13], [15] for the alternating groups, the sporadic simple groups, the Tits group and the simple exceptional groups of Lie type. We note that abelian groups are not determined by the structure of their complex group algebras.…”

Some Extensions of PSL(2,p2) are Uniquely Determined by Their Complex Group Algebras

“…It was conjectured that all nonabelian simple groups are uniquely determined by the structure of their complex group algebras. This conjecture was verified in [13,16,19,20] for the alternating groups, the sporadic simple groups, the Tits group, and the simple groups of exceptional Lie type. We note that abelian groups are not determined by the structure of their complex group algebras.…”

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

“…It was conjectured that all nonabelian simple groups are uniquely determined by the structure of their complex group algebras. This conjecture was verified in [22,23] for the alternating groups, the sporadic simple groups, the Tits group and the simple exceptional groups of Lie type. As pointed out by the referee, a sketch for the case of alternating and sporadic simple groups was already given in [12] and the case for the alternating groups has been worked out in [17].…”