NSE characterization of projective special linear group $L_5(2)$

Abstract

Let $G$ be a group and $\omega (G)$ be the set of element orders of $G$. Let $k\in \omega (G)$ and $s_k$ be the number of elements of order $k$ in $G$. Let nse $(G)=\left\{s_k\right|k\in \omega (G)\}$. In Khatami et al. and Liu, $L_3(2)$ and $L_3(4)$ are uniquely determined by nse $(G)$. In this paper, we prove that if $G$ is a group such that nse $(G)=nse(L_5(2)$), then $G\cong L_5(2)$.