Modules over nilpotent groups of finite rank
type: Item
instance of: scholarly article
title: Modules over nilpotent groups of finite rank
zbMATH Open document ID: 0604.20037
DOI: 10.1007/BF01978845
author: D. I. Zajtsev
author: A. V. Tushev
author: L. A. Kurdachenko
published in: Algebra and Logic
publication date: 1985-01-01T00:00:00Z
review text: This paper arises from the study of various minimality conditions on infinite soluble groups. As usual, this leads to interesting questions about modules over group rings. In this case, the groups in question are nilpotent. Most of the work in the paper is module theoretic, and the results obtained are applied to obtain information about weak minimality conditions. The minimality condition of principal interest here is Min-\(\infty\)-n, a weak version of Min-n, the minimal condition for normal subgroups. For an account of locally soluble groups with Min-n, see \textit{J. S. Wilson} [Proc. Miniconf., Theory of Groups, Canberra 1975, Lect. Notes Math. 573, 130-142 (1977; Zbl 0361.20041)]. For the moment we note that soluble groups with Min-n are periodic (Baer), metabelian ones are countable (McDougall), while soluble groups of derived length three need not be countable (the reviewer). The weak minimality conditions Min-\(\infty\) and Min-\(\infty\)-n are defined as follows. A group G satisfies Min-\(\infty\) (resp. Min-\(\infty\)- n) if in any descending chain \(G_ 1\geq G_ 2\geq..\). of subgroups (resp. normal subgroups) of G, the indices \(| G_ i:G_{i+1}|\) are all finite beyond some point. These conditions have received some attention recently. Analogously to the above, the following results are obtained. Every metabelian group satisfying Min-\(\infty\)-n, is countable (Corollary to Theorem 4.2). Theorem 4.3: For torsion-free metanilpotent groups, the conditions Min-\(\infty\)-n and Min-\(\infty\) coincide, and hence torsionfree metabelian groups with Min-\(\infty\)-n are minimax. Theorem 4.4, which has a fairly straightforward self-contained proof, states that for periodic soluble groups, Min-\(\infty\)-n and Min-n are equivalent. With the exception of Theorem 4.4, these results are deduced from module theoretic theorems. These deal with JG-modules, where J is \({\mathbb{Z}}\) or the group ring \(F<t>\) of an infinite cyclic group over a finite field F, and G is …
reviewed by: Brian Hartley
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20F22
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 3975210
zbMATH Keywords: minimality conditions on infinite soluble groups
zbMATH Keywords: modules over group rings
zbMATH Keywords: weak minimality conditions
zbMATH Keywords: minimal condition for normal subgroups
zbMATH Keywords: metanilpotent groups
zbMATH Keywords: metabelian groups
zbMATH Keywords: periodic soluble groups
zbMATH Keywords: nilpotent group of finite free rank
zbMATH Keywords: central series
MaRDI profile type: Publication
cites work: Finiteness Conditions for Soluble Groups
cites work: On the Finiteness of Certain Soluble Groups
cites work: Soluble groups which are products of nilpotent minmax groups
cites work: Q5182382
cites work: Q4745979
cites work: Q3859990
cites work: Uncountable Artinian Modules and Uncountable Soluble Groups Satisfying Min-n
cites work: Q3218297
full work available at URL: https://doi.org/10.1007/bf01978845
OpenAlex ID: W1965111613
description: scientific article; zbMATH DE number 3975210