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

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

2011-02-22

arxiv.org/abs/1102.4427v1

Mathematics

Group Theory

18 pages

Scientific paper

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$ forgetting multiplicities, and let $\textrm{X}_1(G)$ 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 $\textrm{cd}(S)\subseteq \textrm{cd}(H)$ then $S$ must be isomorphic to $H.$ As a consequence, we show that if $G$ is a finite group with $\textrm{X}_1(G)\subseteq \textrm{X}_1(H)$ 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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