Nilpotent groups with every finite homomorphic image cyclic
type: Item
instance of: scholarly article
title: Nilpotent groups with every finite homomorphic image cyclic
zbMATH Open document ID: 0633.20023
DOI: 10.1007/BF01200221
author: Luise-Charlotte Kappe
author: Robert F. Chamberlain
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: The authors call a group an FHC-group if every finite homomorphic image of it is cyclic, and an AFHC-group if it is, moreover, abelian. Arbitrary FHC-groups can be arbitrarily complicated, as they include all infinite simple groups, and also some ingenious examples made by \textit{G. Baumslag} [J. Aust. Math. Soc. 10, 497-498 (1969; Zbl 0214.274); Bull. Aust. Math. Soc. 6, 313-314 (1972; Zbl 0226.20024)] of one-relator and of locally free FHC-groups. A characterization of AFHC-groups flows from a note by \textit{R. Gilmer} [Am. Math. Mon. 81, 1036-1037 (1974)]. The authors now study nilpotent FHC-groups, which in this review will be called NFHC- groups. Only a selection of the numerous results can be quoted here. If the derived group of an FHC-group is finite, then the group is abelian (or, in the reviewer's notation, \(FHC\cap FD=AFHC)\). If a group is FHC, then the factor group by its derived group is AFHC, and all further lower central factor groups are divisible; and for nilpotent groups the converse is also true. The torsion subgroup of an NFHC-group is AFHC. Examples in the final section of the paper show that the upper central factors of NFHC-groups need not be FHC-groups; in fact they can be quite arbitrary abelian groups. Throughout the paper there is a clear discussion of its relation to known results.
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 4029824
zbMATH Keywords: finite homomorphic image
zbMATH Keywords: FHC-groups
zbMATH Keywords: locally free FHC-groups
zbMATH Keywords: AFHC- groups
zbMATH Keywords: nilpotent FHC-groups
zbMATH Keywords: NFHC-groups
zbMATH Keywords: derived group
zbMATH Keywords: lower central factor groups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: torsion subgroup
zbMATH Keywords: upper central factors
reviewed by: B. H. Neumann
MaRDI profile type: Publication
cites work: Abelian groups without elements of finite order
cites work: A non-cyclic one-relator group all of whose finite quotients are cyclic
cites work: A non-cyclic, locally free, free-by-cyclic group all of whose finite factor groups are cyclic
cites work: Q5846339
cites work: Q3265704
cites work: Q5682962
cites work: The classification of certain classes of torsion free Abelian groups
cites work: Groups Covered By Permutable Subsets
cites work: Q3941589
cites work: Groups That are the Union of Finitely Many Proper Subgroups
Wikidata QID: Q122192920
full work available at URL: https://doi.org/10.1007/bf01200221
OpenAlex ID: W1969092223
description: scientific article; zbMATH DE number 4029824