Fixed point ratios in actions of finite classical groups. I.
type: Item
instance of: scholarly article
title: Fixed point ratios in actions of finite classical groups. I.
zbMATH Open document ID: 1128.20003
author: Timothy C. Burness
published in: Journal of Algebra
publication date: 2007-04-11T00:00:00Z
review text: Let \(G\) be a finite group acting on a finite set \(\Omega\), and let \(C_\Omega(x)\) denote the set of points in \(\Omega\) fixed by \(x\in G\). Define the `fixed point ratio of \(x\)' to be the proportion of points in \(\Omega\) fixed by \(x\); i.e. \(\text{fpr}(x)=|C_\Omega(x)|/|\Omega|\). This paper is the first in a series of four papers which study this ratio for finite classical groups [see also \textit{T. C. Burness}, part II, ibid. 309, No. 1, 80-138 (2007; see the following review Zbl 1128.20004), part III, ibid. 314, No. 2, 693-748 (2007; Zbl 1133.20003), part IV, ibid. 314, No. 2, 749-788 (2007; Zbl 1133.20004)]. The strategy is to reduce the problem to a study of conjugacy classes in \(G\), using the simple observation that if the action of \(G\) is transitive, then \(\text{fpr}(x)=|x^G \cap H|/|x^G|\), where \(H\) is the stabiliser of some point in \(\Omega\). This naturally leads to a consideration of the maximal subgroups in a classical group, famously classified by \textit{M. Aschbacher} [Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)]. Certain of the subgroups in Aschbacher's classification can be naturally called `subspace subgroups'; roughly speaking, these arise as stabilisers of subspaces in the natural module for \(G\) (e.g., parabolic subgroups). By extension, a transitive \(G\)-action is called a `subspace action' if the stabiliser of a point of \(\Omega\) is a subspace subgroup; such actions give rise to relatively large fixed point ratios. The aim of this series of papers is to provide concrete bounds on the fixed point ratio for `non-subspace' actions. The main result says that, with a small number of exceptions (which are also dealt with), if \(G\) is a finite almost simple classical group and \(\Omega\) is a faithful transitive non-subspace \(G\)-set, then \(\text{fpr}(x)\lesssim |x^G|^{-\frac 12}\) for all elements \(x\in G\) of prime order. The proof of the theorem is contained in the later papers, which run through each of the rel…
reviewed by: Michael Bate
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5141710
zbMATH Keywords: finite classical groups
zbMATH Keywords: fixed point ratios
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: monodromy groups
zbMATH Keywords: permutation representations
zbMATH Keywords: finite almost simple groups
zbMATH Keywords: maximal subgroups of classical groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.05.024
OpenAlex ID: W2099901768
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DOI: 10.1016/J.JALGEBRA.2006.05.024
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description: scientific article; zbMATH DE number 5141710