A new characterization of sporadic Higman-Sims and Held groups

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 \( \mathrm{nse}(G) = \{s_k \mid k \in \omega(G)\} \). 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 \( G \cong M \).