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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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