A 2-local characterization of the Higman–Sims group

✍ Scribed by Michael Aschbacher

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 187 KB
• Volume: 260
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(02)00619-1

✦ Synopsis

This is one in a series of papers providing simplified, modern, computer free treatments of the existence and uniqueness of the sporadic groups, and of the normalizers of subgroups of prime order and Sylow subgroups in these groups. Such information is the minimum necessary for purposes of the Classification of the finite simple groups. The series also seeks to avoid appeals to references other than basic texts like [FGT,SG], or other papers in the series which operate under the same constraints.

Here we treat the Higman-Sims group HS. Define HS to be the rank 3 permutation group constructed by D. Higman and C. Sims in [HS]. In the language of Section 2, HS is a Higman-Sims rank 3 group; conversely (essentially) by a result of Wales in [W], up to isomorphism there is at most one such permutation group.

To identify HS in the context of the Classification, one needs a 2-local characterization of HS as an abstract group rather than a permutation group. The characterization given here is one chosen for the purposes of [AS], where the quasithin groups of even characteristic are classified. This is the place in the Classification where HS appears.

Define a finite group G to be of type HS if there exists an involution z in G and

4 , and M/O 2 (M) ∼ = L 3 (2).

We prove:

Theorem 1. Each group of type HS is isomorphic to HS.