Monatshefte für Mathematik

, Volume 166, Issue 3–4, pp 559–577 | Cite as

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 \({{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}}\) 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.


Character degrees Simple exceptional group 

Mathematics Subject Classification (2000)

Primary 20C15 20D05 


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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of KwaZulu-NatalPietermaritzburgSouth Africa

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