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  • Print publication year: 2016
  • Online publication date: December 2015

6 - Low-dimensional classical groups

Summary

In this final chapter we use our earlier work to determine precise results on the r-elusivity of primitive actions of the low-dimensional classical groups. In particular, we prove Theorem 1.5.5.

Introduction

As before, let G ≤ Sym(Ω) be a primitive almost simple classical group over Fq with socle T and point stabiliser H. Recall that H is a maximal subgroup of G and let n be the dimension of the natural T-module V. As described in Section 2.6, Aschbacher's theorem [3] implies that either H is a geometric subgroup in one of the eight collections, or H is a non-geometric subgroup with a simple socle that acts irreducibly on V (we use to denote the latter subgroup collection). Also recall that a small additional collection of novelty subgroups (denoted by N) arises if T = Sp4(q)' (with p = 2) or PΩ8+ (q).

In general, it is very difficult to give a complete description of the maximal subgroups of G, and in particular those in the collection. However, the maximal subgroups in the low-dimensional classical groups with n ≤ 12 are determined in the recent book [13] by Bray, Holt and Roney-Dougal. By applying our earlier work in Chapters 4 and 5 on geometric actions, we can use the information in [13] to obtain precise results on the r-elusivity of all primitive actions of the low-dimensional classical groups (including non-geometric actions).We anticipate that this sort of information may be useful in applications.

The main result of this chapter is Theorem 6.1.1 below. For convenience we will assume that n ≤ 5, although it would be feasible to extend the analysis to higher dimensions.

Theorem 6.1.1Let G ≤ Sym(Ω) be a primitive almost simple classical group with socle T and point stabiliser H, where

Let r be a prime. Then T is r-elusive if and only if (G,H, r) is one of the cases recorded in Tables 6.4.1–6.4.8.

The proof of Theorem 6.1.1 for geometric actions is essentially a straightforward application of our earlier work in Chapters 4 and 5. Now assume H is a maximal non-geometric subgroup of G, so H is almost simple, with socle S say.