Subgroup separability of certain HNN extensions of finitely generated Abelian groups
type: Item
instance of: scholarly article
title: Subgroup separability of certain HNN extensions of finitely generated Abelian groups
zbMATH Open document ID: 0901.20015
author: P. C. Wong
published in: Rocky Mountain Journal of Mathematics
publication date: 1998-09-13T00:00:00Z
full work available at URL: CONT27-1.html
review text: A group \(G\) is called subgroup separable if, for each finitely generated subgroup \(M\) and for each \(x\not\in M\), there exists a normal subgroup \(N\) of finite index in \(G\) such that \(x\not\in MN\). In this paper, the author gives a characterization of certain HNN-extensions to be subgroup separable. More precisely, he proves the following theorem: Let \(G=\langle t,K;\;t^{-1}At=B,\;\phi\rangle\) be an HNN-extension, where \(K\) is a finitely generated Abelian group and \(A\) and \(B\) have finite index in \(K\). Then the following are equivalent: (i) \(G\) is subgroup separable; (ii) Either \(K=A=B\) or there exists a subgroup \(H\) of finite index in \(K\) and \(H\) is normal in \(G\); (iii) There exists a finitely generated Abelian group \(X\) such that \(K\) is a subgroup of finite index in \(X\) and an automorphism \(\overline\phi\in\Aut X\) with \(\overline\phi|_A=\phi\). For the proof, the author uses also three lemmas from a paper of the reviewer and \textit{E. Raptis} and \textit{D. Varsos} [Arch. Math. 53, No. 2, 121-125 (1989; Zbl 0651.20060)]. Note that Lemma 7 is Theorem 2 of the above mentioned paper, phrased slightly differently and with essentially the same proof. (Also submitted to MR).
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1051541
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: finitely generated groups
zbMATH Keywords: HNN-extensions
MaRDI profile type: Publication
OpenAlex ID: W4239422644
cites work: Q5679027
cites work: Residual finiteness and Hopficity of certain HNN extensions
cites work: A characterization of residually finite HNN-extensions of finitely generated Abelian groups
cites work: Extending isomorphisms to automorphisms
cites work: Residually finite hnn extensions
cites work: Coset Representations in Free Groups
cites work: On homomorphisms onto finite groups
cites work: Subgroup separability of certain HNN extensions
DOI: 10.1216/RMJM/1181071967
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description: scientific article; zbMATH DE number 1051541