On transitive extensions of the Higman-Sims group (Q1843835)

This paper deals with the question of existence of rank-3 transitive extensions of the Higman-Sims simple group. The following two results are obtained: Theorem 1. There exist no rank-3 transitive extensions of the Higman-Sims group in which the two nontrivial suborbits are primitive. Theorem 2. There exist no rank-3 transitive extensions of the Higman-Sims group with subdegrees \(\leq 25,000\). The proofs of the two theorems are based on the results of \textit{D. G. Higman} on rank-3 permutation groups [Math. Z. 86, 145--156 (1964; Zbl 0122.03205)], the characterization of the Higman-Sims group by its Svlow-2 subgroup by \textit{D. Gorenstein} and \textit{M. E. Harris} [J. Algebra 24, 565--590 (1973; Zbl 0256.20013)], and the knowledge of the primitive representations of the Higman-Sims group from the author's thesis [see also Bull. Am. Math. Soc. 77, 535--539 (1971; Zbl 0226.20012)]. Since the appearance of the paper the author has improved the result to the following: Theorem. There exist no rank-3 transitive extensions of the Higman-Sims simple group.