A family of solvable linear groups that do not satisfy the maximal condition on normal subgroups
type: Item
instance of: scholarly article
title: A family of solvable linear groups that do not satisfy the maximal condition on normal subgroups
zbMATH Open document ID: 0613.20020
DOI: 10.1007/BF01157997
author: A. N. Lyul'ko
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: For integers \(n\geq 3\), \(m\geq 2\) let \(G_{n,m}\) be the group of all upper triangular matrices of degree n having the form \[ \begin{pmatrix} 1&*&...&*&* \\ &*&...&*&* \\ && \ddots & \vdots& \vdots \\ &&&*&* \\ &&&&1 \end{pmatrix} \] over the ring \({\mathbb{Z}}[\frac{1}{m}]\), with diagonal elements of the form \(m^ k\), \(k\in {\mathbb{Z}}\). \textit{H. Abels} has shown [Lond. Math. Soc. Lect. Note Ser. 36, 205-211 (1979; Zbl 0422.20026)] that \(G_{4,m}\) is finitely presented but has infinitely generated centre. In the present paper it is shown that the same holds for all groups \(G_{n,m}\) \(n\geq 4\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20F16
zbMATH DE Number: 3991767
zbMATH Keywords: finitely presented soluble groups
zbMATH Keywords: HNN-extensions
zbMATH Keywords: upper triangular matrices
zbMATH Keywords: infinitely generated centre
reviewed by: Robert Bieri
MaRDI profile type: Publication
cites work: Q3855376
cites work: Q4745979
cites work: Q3856933
full work available at URL: https://doi.org/10.1007/bf01157997
OpenAlex ID: W2023662166
description: scientific article; zbMATH DE number 3991767