On the space of subgroups of a compact group. I
type: Item
instance of: scholarly article
title: On the space of subgroups of a compact group. I
zbMATH Open document ID: 1169.22003
author: S. Fisher
author: Paul Gartside
published in: Topology and its Applications
publication date: 2009-04-03T00:00:00Z
review text: [This paper is preparatory to, and should be read prior to, the companion paper ``On the space of subgroups of a compact group. II'', although inexplicably ``I'' follows ``II'' in the printed issue of Topology and its Applications.] Given a compact topological group \(G\) (always Hausdorff here), the authors study in detail these sets of subspaces of \(G\), all in the usual Vietoris topology: \(\mathcal S(G)\), the set of closed subgroups of \(G\); \(\mathcal C(G)\), the set of cosets \(xH\), with \(x\in G\), \(H\in\mathcal S(G)\); and \(\mathcal N(G)\), the set of closed, normal subgroups of \(G\). Among the authors' several results are these: {\parindent=7mm \begin{itemize}\item[(1)] \(\mathcal S(G)\), \(\mathcal C(G)\), and \(\mathcal N(G)\) are compact Hausdorff spaces, and \(\mathcal S\), \(\mathcal C\) and \(\mathcal N\) are covariant functors from the category of compact Hausdorff groups to the category of compact Hausdorff spaces. \item[(2)] \(\mathcal S(G)\) is a retract of \(\mathcal C(G)\). \item[(3)] \(\mathcal S(G)\) is naturally homeomorphic to the orbit space \(\mathcal C(G)/G\). \item[(4)] The topological characters \(\chi(I,\mathcal S(G))\) and \(\chi(1_G,G)\) are equal, where \(I=\{1_G\}\in\mathcal S(G)\). \item[(5)] For \(H\in\mathcal S(G)\), the topology of \(\mathcal S(H)\) is the topology inherited from \(\mathcal S(G)\supseteq\mathcal S(H)\). \item[(6)] For compact \(G_1\), \(G_2\), the map \(h:\mathcal S(G_1)\times\mathcal S(G_1)\rightarrow\mathcal S(G_1\times G_2)\) defined by \(h(H_1,H_2)=H_1\times H_2\) is surjective (hence, a homeomorphism) iff \(G_1\) and \(G_2\) are profinite (=~totally disconnected) and \(\Sigma(G_1)\cap\Sigma(G_2)=\emptyset\); here, \(\Sigma(G)\) is the set of primes \(p\) for which \(G\) contains a topological copy of either \(\mathbb Z_p\) or \(C_{p^n}\) for some \(n\geq1\). \end{itemize}}
reviewed by: W. W. Comfort
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 22C05
zbMATH DE Number: 5539509
zbMATH Keywords: topological group
zbMATH Keywords: closed subgroup
zbMATH Keywords: hyperspace
MaRDI profile type: Publication
full work available at URL: j.topol.2008.11.005
OpenAlex ID: W4210683843
cites work: Q3741814
cites work: Q4692983
cites work: Q3994935
cites work: On the space of subgroups of a compact group. II
cites work: Counting the closed subgroups of profinite groups
cites work: Classifying spaces of subgroups of profinite groups
cites work: Q5723435
cites work: Ein Darstellungssatz für kompakte, total unzusammenhängende Gruppen
cites work: Topologies on Spaces of Subsets
cites work: Shorter Notes: Profinite Groups are Galois Groups
cites work: Q4217279
cites work: On the Completeness of the Space of Compact Subsets
DOI: 10.1016/J.TOPOL.2008.11.005
Recommended article: On the space of subgroups of a compact group. II
Recommended article: Compact subspaces in the space of subgroups of a topological group
Recommended article: On the compact space of closed subgroups of locally compact groups
Recommended article: A note on compact subgroups of topological groups
Recommended article: On subgroups of finite index in compact Hausdorff groups.
Recommended article: Q5442226
Recommended article: Topological groups with \(\sigma\)-compact space of subgroups
Recommended article: On \(L\)-subgroups of locally compact groups
Recommended article: Compact Subgroups of Lie Groups and Locally Compact Groups
Recommended article: Q3789804
description: scientific article; zbMATH DE number 5539509