On base sizes for algebraic groups
type: Item
instance of: scholarly article
title: On base sizes for algebraic groups
zbMATH Open document ID: 1445.20002
DOI: 10.4171/JEMS/718
author: Timothy C. Burness
author: Jan Saxl
author: Robert Guralnick
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2017-07-31T00:00:00Z
review text: The authors outline their investigations: ``Let \(G\) be a (closed) connected affine algebraic group over an algebraically closed field \(K\) of characteristic \(p \geq 0\). Let \(\Omega\) be a faithful transitive \(G\)-variety with point stabilizer \(H\), so we may identify \(\Omega\) with the coset variety \(G/H\). We define three base-related measures that arise naturally in this context: (i) The exact base size, denoted \(b(G, H )\), is the smallest integer \(c\) such that \(\Omega\) contains \(c\) points with trivial pointwise stabilizer. (ii) The connected base size, denoted \(b^0(G,H)\), is the smallest integer \(c\) such that \(\Omega\) contains \(c\) points whose pointwise stabilizer has trivial connected component, i.e. the pointwise stabilizer is finite. (iii) The generic base size, denoted \(b^1(G,H)\), is the smallest integer \(c\) such that the product variety \(\Omega^c = \Omega\times \cdots \times \Omega\) (\(c\) factors) contains a non-empty open subvariety \(\Lambda\) and every \(c\)-tuple in \(\Lambda\) is a base for \(G\). Evidently, we have \[b^0(G,H) \leq b(G,H) \leq b^1(G,H).\] Our ultimate goal is to determine these base-related measures for all simple algebraic groups \(G\) and all closed maximal subgroups \(H\) of \(G\) (that is, for all primitive actions of simple algebraic groups). Indeed, we essentially achieve this goal by computing these quantities in almost every case. In the handful of exceptional cases, we give a very narrow range for the possible values.'' The authors give a list of 73 references. They refer extensively to previous research. They include a number of lists of results such as: some subspace actions; values of \(b\) in Theorem 5(ii); \(G\) exceptional group, \(H\) parabolic subgroup; values of \(b\) in Theorem 7(ii); involutions inverting maximal tori, \(p \neq 2\); non-subspace involution-type subgroups; values of \(b\) in Theorem 3.13(ii); the \(\mathcal{C}_i \) collections; some maximal non-parabolic subgroups of e…
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G41
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 6755103
zbMATH Keywords: base size
zbMATH Keywords: simple algebraic groups
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: generic stabilizer
MaRDI profile type: Publication
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On base sizes for algebraic groups (full work available at URL)
full work available at URL: https://doi.org/10.4171/jems/718

On base sizes for algebraic groups (OpenAlex ID)
OpenAlex ID: W14581337