Uncountable groups with restrictions on subgroups of large cardinality.
type: Item
instance of: scholarly article
title: Uncountable groups with restrictions on subgroups of large cardinality.
zbMATH Open document ID: 1333.20037
author: F. de Giovanni
author: M. Trombetti
published in: Journal of Algebra
publication date: 2015-12-07T00:00:00Z
review text: The authors look for group classes \(\mathbf P\) and (possibly) additional conditions for groups \(G\) of cardinality \(\aleph\) such that \(G\) is a \(\mathbf P\)-group whenever all proper equicardinal subgroups are \(\mathbf P\)-groups. Here with \(\aleph\) is meant an uncountable cardinality. If \(G\) has no simple homomorphic images of cardinality \(\aleph\) and \(\aleph\) is regular, \(G\) is an FC-group if all equicardinal proper subgroups are (Theorem 3.2). If the locally graded group \(G\) has no simple homomorphic images of cardinality \(\aleph\) and \(\aleph\) is regular, \(G\) is nilpotent-by-finite if all equicardinal subgroups are (Theorem 4.6). The absence of equicardinal simple homomorphic images is needed because of the existence of groups with all proper subgroups of strictly lower cardinality (Jónsson groups).
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 03E75
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 6517045
zbMATH Keywords: uncountable groups
zbMATH Keywords: equicardinal subgroups
zbMATH Keywords: FC-groups
zbMATH Keywords: nilpotent-by-finite groups
reviewed by: Hermann Heineken
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.10.002
OpenAlex ID: W2117938457
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DOI: 10.1016/J.JALGEBRA.2015.10.002
description: scientific article; zbMATH DE number 6517045