Fixed point ratios in actions of finite classical groups. I. (English) Zbl 1128.20003
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 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 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 relevant Aschbacher classes in turn.
In the current paper the author introduces the main result and describes how it can be applied to the study of minimal bases for primitive permutation groups. He also outlines how the main result may be useful in classifying those primitive permutation groups which arise as the monodromy groups of a branched covering \(\varphi\colon X\to\mathbb{P}^1\mathbb{C}\), where \(X\) is a compact connected Riemann surface.
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 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 relevant Aschbacher classes in turn.
In the current paper the author introduces the main result and describes how it can be applied to the study of minimal bases for primitive permutation groups. He also outlines how the main result may be useful in classifying those primitive permutation groups which arise as the monodromy groups of a branched covering \(\varphi\colon X\to\mathbb{P}^1\mathbb{C}\), where \(X\) is a compact connected Riemann surface.
Reviewer: Michael Bate (Oxford)
MSC:
| 20B15 | Primitive groups |
| 20G40 | Linear algebraic groups over finite fields |
| 20G05 | Representation theory for linear algebraic groups |
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
| 14L30 | Group actions on varieties or schemes (quotients) |
Keywords:
finite classical groups; fixed point ratios; primitive permutation groups; monodromy groups; permutation representations; finite almost simple groups; maximal subgroups of classical groupsReferences:
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