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

2011/02/22 by Tong-Viet, Hung P.
#20C15 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1102.4427

Abstract

Let G be a finite group. Denote by \textrmIrr(G) the set of all irreducible complex characters of G. Let \textrmcd(G)=\χ(1) | χ∈ \textrmIrr(G)\ be the set of all irreducible complex character degrees of G forgetting multiplicities, and let \textrmX₁(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 \textrmcd(S)⊆ \textrmcd(H) then S must be isomorphic to H. As a consequence, we show that if G is a finite group with \textrmX₁(G)⊆ \textrmX₁(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.

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