On finitely generated subgroups of some soluble groups
type: Item
instance of: scholarly article
title: On finitely generated subgroups of some soluble groups
zbMATH Open document ID: 0603.20032
DOI: 10.1007/BF01769225
author: P. Longobardi
author: M. Maj
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathcal H}\) denote the class of groups in which the intersection of two finitely generated subgroups is finitely generated, and \({\mathcal K}\) the class of groups G in which each finitely generated subgroup is polycyclic or of finite index in G. Among the many interesting results in this paper are the following: A. If G is nilpotent-by-abelian and has max-n, then the following are equivalent: (i) \(G\in {\mathcal H}\) (ii) \(G\in {\mathcal K}\) (iii) G'' is polycyclic and G/G''\(\in {\mathcal H}\). B. There exists a finitely generated, nilpotent-by-abelian, \({\mathcal H}\)-group G which is not in \({\mathcal K}\) and has G'' non-polycyclic. C. For soluble groups of finite rank the properties \({\mathcal H}\) and \({\mathcal K}\) coincide. D. If G is finitely generated, nilpotent-by-abelian, then G is in \({\mathcal K}\) if and only if every section of G is in \({\mathcal H}\).
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 3973156
zbMATH Keywords: intersection of two finitely generated subgroups
zbMATH Keywords: nilpotent-by-abelian
zbMATH Keywords: max-n
zbMATH Keywords: polycyclic
zbMATH Keywords: soluble groups of finite rank
reviewed by: D. J. McCaughan
MaRDI profile type: Publication
cites work: On the Intersection of Finitely Generated Free Groups
cites work: Q3342725
cites work: Intersections of finitely generated subgroups in metabelian groups
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cites work: Q5654235
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description: scientific article; zbMATH DE number 3973156