Recognition of $$\text {PSL}(2,p)$$ PSL ( 2 , p ) by order and some information on its character degrees where $$p$$ p is a prime

Abstract

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.