A characterization of the linear groups L 2(p)

Let G be a finite group and pi (e) (G) be the set of element orders of G. Let k a pi (e) (G) and m (k) be the number of elements of order k in G. Set nse(G):= {m (k) : k a pi (e) (G)}. In fact nse(G) is the set of sizes of elements with the same order in G. In this paper, by nse(G) and order, we give a new characterization of finite projective special linear groups L (2)(p) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that |G| = |L (2)(p)| and nse(G) consists of 1, p (2) - 1, p(p + E >)/2 and some numbers divisible by 2p, where p is a prime greater than 3 with p a parts per thousand 1 modulo 4, then G a parts per thousand... L (2)(p).