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Recognition of \(\text{PSL}(2,p)\) by order and some information on its character degrees where \(p\) is a prime. (English) Zbl 1304.20042
Summary: Let \(G\) be a finite group and \(\text{cd}(G)\) be the set of irreducible character degrees of \(G\). In this paper we prove that if \(p\) is a prime number, then the simple group \(\text{PSL}(2,p)\) is uniquely determined by its order and some information about its character degrees. In fact we prove that if \(G\) is a finite group such that (i) \(|G|=|\text{PSL}(2,p)|\), (ii) \(p\in\text{cd}(G)\), (iii) \(\text{cd}(G)\) has an even integer, and (iv) there does not exist any element \(a\in\text{cd}(G)\) such that \(2p\mid a\), then \(G\cong\text{PSL}(2,p)\). As a consequence of our result we get that \(\text{PSL}(2,p)\) is uniquely determined by its order and the largest and the second largest character degrees.

MSC:
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20C15 Ordinary representations and characters
20D06 Simple groups: alternating groups and groups of Lie type
20C33 Representations of finite groups of Lie type
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