Characterization of $L_2(16)$ by $\tau_e(L_2(16))$

Abstract:

      Let $G$ be a group and $\pi_e(G)$ the set of element orders of $G$. Let $k\in \pi_e(G)$ and $m_k$ be the number of elements of order $k$ in $G$. Let $\tau_e(G)=\{m_k |k\in \pi_e(G)\}$. In this paper, we prove that $L_2(16)$ is recognizable by $\tau_e(L_2(16))$. In other words, we prove that if $G$ is a group such that $\tau_e(G)=\tau_e(L_2(16))=\{1, 255, 272, 544, 1088, 1920\}$, then $G$ is isomorphic to $L_2(16)$.