Residual properties of HNN-extensions with base group an Abelian group
type: Item
instance of: scholarly article
title: Residual properties of HNN-extensions with base group an Abelian group
zbMATH Open document ID: 0676.20012
DOI: 10.1016/0022-4049(89)90098-4
author: E. Raptis
author: Dimitrios Varsos
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: As the title indicates, this paper deals with residual properties of the HNN-extension \(G=<t,K|\) \(t^{-1}At=B,\phi >\), where K is a finitely generated abelian group. It is proved that G is residually solvable. The authors give also necessary conditions for the residual nilpotence of G. In the special case where \(K=A\) or \(K=B\), or in the special case where \(A=B\), they give a characterization for G to be residually nilpotent. More precisely they define the map (endomorphism) f, by \(\alpha f=\alpha^{-1}(\alpha \phi)\) and prove that if \(K=A\), then G is residually nilpotent iff the only f-invariant subgroup of A is trivial, if \(A=B\) then G is residually nilpotent iff the only f-invariant subgroup is trivial and the torsion part of K/A is a finite p-group, for a prime p.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 4107080
zbMATH Keywords: HNN-extension
zbMATH Keywords: finitely generated abelian group
zbMATH Keywords: residually solvable
zbMATH Keywords: residually nilpotent
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90098-4
OpenAlex ID: W1975160317
cites work: A characterization of residually finite HNN-extensions of finitely generated Abelian groups
cites work: Extending isomorphisms to automorphisms
cites work: Residual Properties of Infinite Soluble Groups
cites work: Subgroups of HNN Groups and Groups with one Defining Relation
Wikidata QID: Q127764020
description: scientific article; zbMATH DE number 4107080