Characterization of projective general linear groups

Document Type: Research Paper

Author

Farhangian University, Shariati Mazandaran, Iran

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 $PGL$_{2}(q)$‎.

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