Simple exceptional groups of Lie type are determined by their character degrees |
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Authors: | Hung P Tong-Viet |
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Institution: | 1. School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa
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Abstract: | 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. |
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