Permutability in uncountable groups
type: Item
instance of: scholarly article
title: Permutability in uncountable groups
zbMATH Open document ID: 1464.20007
author: Xianqiang Yang
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2019-01-14T00:00:00Z
review text: A group is \textit{quasihamiltonian} if each of its subgroups is permutable (or quasinormal), which means that \(HK=KH\) for all subgroups \(H\) and \(K\) of the group. Permutable subgroups were introduced by \textit{Ø. Ore} [Duke Math. J. 3, 149--174 (1937; Zbl 0016.35103)] and \textit{K. Iwasawa} [Jpn. J. Math. 18, 709--728 (1943; Zbl 0061.02504)] completely described quasihamiltonian groups in 1943. Recently, groups whose uncountable subgroups are normal have been studied (see [\textit{M. de Falco} et al., Adv. Group Theory Appl. 3, 13--29 (2017; Zbl 1390.20031)]) and the goal of this work is to do a similar investigation using permutable subgroups. Examples of \textit{A. Ehrenfeucht} and \textit{V. Faber} [Nederl. Akad. Wet., Proc., Ser. A 75, 202--209 (1972; Zbl 0256.20032)] show that uncountable groups whose uncountable subgroups are permutable are not always quasihamiltonian groups, even in the periodic nilpotent case, and the authors point out that the absence of large (i.e. uncountable) abelian subgroups is decise in this context (Lemma 2.3). In any case, the authors are able to prove that uncountable groups whose large subgroups are permutable have a proper commutator subgroup whenever there are no large homomorphic images (Theorem 2.10) and hence a small (i.e. countable) second commutator subgroup (Corollary 2.11). The study splits in two cases according to which the group is periodic or not, but in both cases under the assumption that there are no large simple sections. In the non-periodic case they manage to obtain a complete characterization (Theorem 3.5) that is similar to that of Iwasawa (see [\textit{R. Schmidt}, Subgroup lattices of groups. Berlin: Walter de Gruyter (1994; Zbl 0843.20003)], Theorem 2.4.11). In the periodic case, they show that such groups contain a large direct factor which is a locally finite \(p\)-group (Theorem 4.1).
reviewed by: Maria Ferrara
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 7001285
zbMATH Keywords: uncountable group
zbMATH Keywords: permutable subgroup
zbMATH Keywords: quasihamiltonian group
MaRDI profile type: Publication
full work available at URL: s10231-018-0730-3
OpenAlex ID: W2794300902
cites work: Q4609057
cites work: Large soluble groups and the control of embedding properties.
cites work: A note on groups of infinite rank with modular subgroup lattice.
cites work: Groups in which every non-Abelian subgroup is permutable.
cites work: Uncountable groups with restrictions on subgroups of large cardinality.
cites work: A note on uncountable groups with modular subgroup lattice
cites work: NILPOTENCY IN UNCOUNTABLE GROUPS
cites work: Groups with all subgroups permutable or of finite rank.
cites work: Q5670800
cites work: On Groups in Which Every Subgroup of Infinite Rank Is Subnormal of Bounded Defect
cites work: Q5675882
cites work: Q4820338
cites work: AN IMBEDDING THEOREM FOR GROUPS AND ITS COROLLARIES
cites work: Q5654236
cites work: Q3879436
cites work: Subgroup lattices of groups
cites work: Permutable subgroups of infinite groups
cites work: Q5559294
DOI: 10.1007/S10231-018-0730-3
description: scientific article; zbMATH DE number 7001285