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Simple exceptional groups of Lie type are determined by their character degrees. (English) Zbl 1255.20006
Let $$\text{cd}(G)$$ denote the set of all irreducible complex character degrees of the group $$G$$. Let $$X_1(G)$$ denote the set of all irreducible complex character degrees of $$G$$ counting multiplicities. In the paper under review the author proves that for $$H$$ a non-Abelian simple exceptional group of Lie type, if $$S$$ is any simple group with $$\text{cd}(S)$$ a subset of $$\text{cd}(H)$$, then $$S$$ is isomorphic to $$H$$ and if $$X_1(G)$$ is a subset of $$X_1(H)$$ for any finite group $$G$$, then $$G$$ is isomorphic to $$H$$. As a consequence if the group algebras $$\mathbb CG$$ and $$\mathbb CH$$ are isomorphic then $$G$$ is isomorphic to $$H$$.

##### MSC:
 20C15 Ordinary representations and characters 20C33 Representations of finite groups of Lie type 20D06 Simple groups: alternating groups and groups of Lie type 20C05 Group rings of finite groups and their modules (group-theoretic aspects)
GAP
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