We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 18 December 2015
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.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
