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

# 5 - Non-subspace actions

## Summary

Introduction

Let G be a primitive almost simple classical group over Fq with point stabiliser H, socle T and natural module V with dimV = n. Write q = pf, where p is a prime. We may assume that n and q satisfy the conditions recorded in Table 3.0.1. As described in Section 2.6, Aschbacher's subgroup structure theorem [3] implies that H belongs to one of ten subgroup collections, denoted by

In the previous chapter we studied derangements of prime order in the case where (the so-called subspace actions) and now we turn our attention to the remaining non-subspace actions of finite classical groups. More precisely, in this chapter we will assume that H belongs to one of the geometric subgroup collections with 2 ≤ i ≤ 8 (see Table 5.1.1 for a brief description of these subgroups). We adopt the precise definition of the collections used by Kleidman and Liebeck [86], and the specific permutation groups with will be handled in Section 5.i. In addition, in Section 5.9 we consider the small collection N of novelty subgroups that arises when T = PΩ8+ (q) or Sp4(q)' (with q even).

As noted in Chapter 1, rather different techniques are needed to handle the almost simple irreducible subgroups comprising the collection, and we will deal with them in a separate paper.

We will continue to adopt the notation introduced in Chapters 2 and 3. In particular, recall that if xG ∩ PGL(V) and is a preimage of x, where and is the algebraic closure of Fq, then we define

where is the subspace of spanned by the vectors (with); see Definition 3.1.4. Observe that v(x) is the codimension of the largest eigenspace of on. Lower bounds on this parameter for xH will play an important role in our study of derangements in this chapter (see Lemmas 5.3.3, 5.4.2, 5.6.3 and 5.7.3, for instance).

Throughout this chapter we will frequently refer to the type of H, which provides an approximate description of the group-theoretic structure of H ∩ PGL(V).

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