Characterization of projective general linear groups

Abstract

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$_{2}(q)|$, where $q$ is odd prime power and nse$(G)= $nse$($PGL$_{2}(q))$, then $G cong $PGL$_