A new characterization of sporadic Higman–Sims and Held groups

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 $\mathrm{nse}(G)=\{s_k|k\in\omega(G)\}$. The projective special linear groups $L_3(4)$ and $L_3(5)$ are uniquely determined by $\mathrm{nse}$. In this paper, we prove that if $G$ is a group such that $\mathrm{nse}(G)=\mathrm{nse}(M)$ where $M$ is a sporadic Higman–Sims or Held group, then $G\cong M$.