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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.

##### MSC:
 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
GAP
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