Normal subgroups of Abels' groups
type: Item
instance of: scholarly article
title: Normal subgroups of Abels' groups
zbMATH Open document ID: 0566.20018
DOI: 10.1007/BF01141931
author: A. N. Lyul'ko
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: \textit{H. Abels} [Lond. Math. Soc. Lect. Note Ser. 36, 205-211 (1979; Zbl 0422.20026)] has shown that for every prime p the group \(T_ p\) of invertible upper triangular matrices of degree 4 over the ring of rationals with denominator a power of p and two (positive) units at the edges of the diagonal is a finitely presented soluble group that does not satisfy the maximal condition for normal subgroups. Here, some further facts about the normal subgroups of \(T_ p\) are obtained. In particular, the maximal normal subgroups of \(T_ p\) are determined and it is shown that \(T_ p\) has countably many normal subgroups, whereas the normal subgroups of the direct square \(T_ p\times T_ p\) have the cardinality of the continuum.
reviewed by: Bernhard Amberg
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 3902894
zbMATH Keywords: upper triangular matrices
zbMATH Keywords: finitely presented soluble group
zbMATH Keywords: maximal condition for normal subgroups
zbMATH Keywords: maximal normal subgroups
MaRDI profile type: Publication
cites work: Q3855376
cites work: Q3856933
cites work: On the cohomology of soluble groups. II
cites work: On the Number of Normal Subgroups of an Infinite Group
cites work: Q4745979
full work available at URL: https://doi.org/10.1007/bf01141931
OpenAlex ID: W2079094518
description: scientific article; zbMATH DE number 3902894