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

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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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Authors and Affiliations

  1. School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa

    Hung P. Tong-Viet

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  1. Hung P. Tong-Viet
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Correspondence to Hung P. Tong-Viet.

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Communicated by John S. Wilson.

The author is supported by a post-doctoral fellowship from the University of KwaZulu-Natal.

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Tong-Viet, H.P. Simple exceptional groups of Lie type are determined by their character degrees. Monatsh Math 166, 559–577 (2012). https://doi.org/10.1007/s00605-011-0301-9

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