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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)
Software:
GAP
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