Closed subgroups of the infinite symmetric group.
type: Item
instance of: scholarly article
title: Closed subgroups of the infinite symmetric group.
zbMATH Open document ID: 1111.20004
author: George M. Bergman
author: S. Shelah
published in: Algebra Universalis
publication date: 2007-01-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0401305
full work available at URL: 5sj0d2v2
review text: If \(\Omega\) is an infinite set and we regard it as a discrete topological space, then the set \(\Omega^\Omega\) of all functions \(\Omega\to\Omega\) becomes a topological space under the function topology (in this topology, a subbasis of open sets is given by the sets \(\{f\in\Omega^\Omega:f(\alpha)=\beta\), \(\alpha,\beta\in\Omega\}\) and the closure of a set \(U\subset\Omega^\Omega\) consists of all maps \(f\) such that, for every finite subset \(\Gamma\subseteq\Omega\), there exists an element of \(U\) agreeing with \(f\) at all members of \(\Gamma\)). Let now \(\Omega\) be a countably infinite set and \(S=\text{Sym}(\Omega)\) the group of all permutations of \(\Omega\). For \(G_1,G_2\leq S\) let us write \(G_1\approx G_2\) if there exists a finite set \(F\subseteq G\) such that \(\langle G_1\cup F\rangle=\langle G_2\cup F\rangle\). Let \(\lambda\) be the least cardinal such that for some finite subset \(\Gamma\subseteq\Omega\), all orbits of \(G_{(\Gamma)}\) (pointwise stabilizer subgroup of \(\Gamma\)), in \(\Omega\) have cardinality \(<\lambda\). In this paper the authors show that the subgroups closed in the function topology on \(S\) lie in precisely four equivalence classes under this relation. -- Which of these classes a closed subgroup \(G\) belongs to depends on which of the following conditions on \(\lambda\) holds: (i) \(\lambda=\aleph_1\), (ii) \(\lambda=\aleph_0\), (iii) \(3\leq\lambda<\aleph_0\), and (iv) \(\lambda=2\).
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 5119633
zbMATH Keywords: full permutation group on countably infinite set
zbMATH Keywords: subgroups closed in function topology
zbMATH Keywords: equivalence relations on subgroups
zbMATH Keywords: cardinalities of orbits of stabilizers of finite sets
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
OpenAlex ID: W2163622252
arXiv ID: math/0401305
DOI: 10.1007/S00012-006-1959-Z
description: scientific article