A solution of the dimension subgroup problem
type: Item
instance of: scholarly article
title: A solution of the dimension subgroup problem
zbMATH Open document ID: 0729.20002
DOI: 10.1016/0021-8693(91)90182-8
author: Narain Gupta
published in: Journal of Algebra
publication date: 1991-01-01T00:00:00Z
review text: Let \({\mathbb{Z}}G\) be the integral group ring of a group G. For \(n\geq 1\) the nth dimension subgroup \(D_ n(G)\) is defined as \(D_ n(G)=G\cap (1+\Delta^ n(G))\) where \(\Delta\) (G) is the augmentation ideal, evidently \(D_ n(\gamma)\geq \gamma_ n(G)\) where \(\gamma_ n(G)\) is the nth term of the lower central series. Since it is now known that, in general, \(D_ n(G)\neq \gamma_ n(G)\) (n\(\geq 4)\), the determination of the factor groups \(D_ n(G)/\gamma_ n(G)\) has become the so-called dimension subgroup problem. For metabelian p-groups the author had extended an earlier result of \textit{J. A. Sjogren} [J. Pure Appl. Algebra 14, 175-194 (1979; Zbl 0395.20019)] to show that \(D_ n(G)=\gamma_ n(G)\) for all \(n\leq 2p-1\) [ibid. 51, 241-249 (1988; Zbl 0649.20004)]. In this paper he establishes that if G is a finite metabelian group then \(D_ n(G)/\gamma_ n(G)\) has exponent which is a power of 2, and that if G is a finitely generated metabelian p-group, p odd, then \(D_ n(G)=\gamma_ n(G)\) for all \(n\geq 1\).
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 4204638
zbMATH Keywords: integral group ring
zbMATH Keywords: dimension subgroup
zbMATH Keywords: augmentation ideal
zbMATH Keywords: lower central series
zbMATH Keywords: metabelian p-groups
MaRDI profile type: Publication
full work available at URL: 0021-8693(91)90182-8
OpenAlex ID: W2028225607
cites work: On the dimension subgroups of metabelian groups
cites work: Dimension subgroups of metabelian p-groups
cites work: Free Group Rings
cites work: Q3658176
cites work: The Dimension Subgroup Conjecture
cites work: Über Beziehungen zwischen höheren Kommutatoren.
cites work: Q5513066
cites work: Q4406395
cites work: On the fourth integer dimension subgroup
cites work: Dimension and lower central subgroups
description: scientific article; zbMATH DE number 4204638