Characterization of projective general linear groups

‎Let G be a finite group and \pi_{e}(G) be the set of element orders of G ‎. ‎Let k \in \pi_{e}(G) and s_{k} be the number of elements of order ‎k in G‎. ‎Set nse(G):=\{ s_{k} | k \in \pi_{e}(G)\}‎. ‎In this paper‎, ‎it‎ ‎is proved if |G|=| PGL_{۲}(q)|‎, ‎where q is odd prime power and nse ‎(G)= nse(PGL_{۲}(q))‎, ‎then G \cong PGL_{۲}(q)‎.

Keywords:

‎Element order‎ , ‎set of the numbers of elements of the same order‎ , ‎projective general linear group