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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 \(H\).

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)
Full Text: DOI arXiv
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