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Alternating and sporadic simple groups are determined by their character degrees. (English) Zbl 1252.20005
Summary: Let \(G\) be a finite group. Denote by \(\text{Irr}(G)\) the set of all irreducible complex characters of \(G\). Let \(\text{cd}(G)\) be the set of all irreducible complex character degrees of \(G\) forgetting multiplicities, that is, \(\text{cd}(G)=\{\chi(1):\chi\in\text{Irr}(G)\}\) and let \(\text{cd}^*(G)\) be the set of all irreducible complex character degrees of \(G\) counting multiplicities. Let \(H\) be an alternating group of degree at least 5, a sporadic simple group or the Tits group. In this paper, we show that if \(G\) is a non-Abelian simple group and \(\text{cd}(G)\subseteq\text{cd}(H)\) then \(G\) must be isomorphic to \(H\). As a consequence, we show that if \(G\) is a finite group with \(\text{cd}^*(G)\subseteq\text{cd}^*(H)\) then \(G\) is isomorphic to \(H\). This gives a positive answer to Question 11.8(a) in The Kourovka notebook. Unsolved problems in group theory [16th ed. (2006; Zbl 1084.20001)] for alternating groups, sporadic simple groups or the Tits group.

20C15 Ordinary representations and characters
20C30 Representations of finite symmetric groups
20C34 Representations of sporadic groups
20D06 Simple groups: alternating groups and groups of Lie type
20D08 Simple groups: sporadic groups
Full Text: DOI
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