Geodesic
A new characterization of sporadic Higman--Sims and Held groups
Eurasian mathematical journal, Tome 5 (2014) no. 3, pp. 102-116

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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 $G\cong M$. 
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     author = {Y. Yang and S. Liu},
     title = {A new characterization of sporadic {Higman--Sims} and {Held} groups},
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}
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Y. Yang; S. Liu. A new characterization of sporadic Higman--Sims and Held groups. Eurasian mathematical journal, Tome 5 (2014) no. 3, pp. 102-116. http://geodesic.mathdoc.fr/item/EMJ_2014_5_3_a6/