CHARACTERIZATION OF PROJECTIVE GENERAL LINEAR GROUPS

Let G be a nite group and e(G) be the set of element orders of G. Let k 2 e(G) and sk be the number of elements of order k in G. Set nse(G):=fskjk 2 e(G)g. In this paper, it is proved if jGj = jPGL2(q)j, where q is odd prime power and nse(G) = nse(PGL2(q)), then G = PGL2(q). nite group, then (j G j ) is denoted by (G ). We denote bye(G ) the set of orders of its elements. It is clear that the sete(G ) is closed and partially ordered by divisibility, and hence it is uniquely determined by (G ), the subset of its maximal elements. The prime graph ( G ) of a group G is dened as a graph with vertex set (G ) in which two distinct primes p , q 2 (G ) are adjacent if G contains an element of order pq . Let t (G ) be the number of connected components of ( G ) and 1; : : : ; t(G) be the connected components of ( G ). If 2 2 (G ), then we always suppose that 2 2 1. Then 1 is called the even component of ( G ) and 2; : : : ; t(G) are called the odd components of ( G ). Let p be a prime. A group G is called a C pp if p 2 (G ) and p is an isolated vertex of the prime graph of G. In the other words, the centralizers of its elements of order p in G are p -groups. Given a