Groups with countably many subgroups.
type: Item
instance of: scholarly article
title: Groups with countably many subgroups.
zbMATH Open document ID: 1342.20030
author: Giovanni Cutolo
author: Howard Smith
published in: Journal of Algebra
publication date: 2016-01-15T00:00:00Z
review text: It is an easy observation that a group is finite precisely when it has only finitely many subgroups. In the very nice paper under review the authors consider groups \(G\) in which the set of all subgroups of \(G\) is countable. The abelian groups with this property are classified in [\textit{S. V. Rychkov} and \textit{A. A. Fomin}, ``Abelian groups with a countable number of subgroups'', Abelevy Gruppy Moduli 10, 99-105 (1991)]. The current paper is mostly concerned with soluble such groups. A group \(G\) is termed a CMS-group if \(\mathcal L(G)\), the set of all subgroups of \(G\) is countable. Of course, all CMS-groups are countable and also groups with the maximum condition are CMS-groups. The class of CMS-groups is closed under taking subgroups and quotients, but \(C_{p^\infty}\times C_{p^\infty}\) (which has \(2^{\aleph_0}\) subgroups) shows that the property is not extension closed. The authors show (Theorem 2.7) that a soluble-by-finite group is CMS if and only it is minimax and has no (subnormal) sections of type \(C_{p^\infty}\times C_{p^\infty}\). They also show that locally (soluble-by-finite) CMS-groups are soluble-by-finite (Theorem 2.12). Interestingly also (Theorem 3.1) they give an example of a nilpotent group with uncountably many subgroups, of which just countably many are abelian. On the other hand, if \(G\) is an infinite soluble-by-finite group then \(|\mathcal L(G)|\) is either \(\aleph_0\) or \(2^{|G|}\). The construction of Theorem 35.2 of \textit{A. Yu. Ol'shanskij} [Geometry of defining relations in groups. Dordrecht: Kluwer Academic Publishers (1991; Zbl 0732.20019)] provides a simple group of cardinality \(\aleph_1\) with exactly \(\aleph_1\) subgroups, all proper subgroups are countable and \(G\) may be arranged to be a \(p\)-group, for a sufficiently large prime \(p\).
reviewed by: Martyn Dixon
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 6530854
zbMATH Keywords: soluble groups
zbMATH Keywords: CMS-groups
zbMATH Keywords: countable sets of subgroups
zbMATH Keywords: Abelian subgroups
zbMATH Keywords: soluble-by-finite groups
zbMATH Keywords: minimax groups
zbMATH Keywords: group with countably many subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.10.007
OpenAlex ID: W2195441172
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DOI: 10.1016/J.JALGEBRA.2015.10.007
description: scientific article; zbMATH DE number 6530854