On modular relation modules and residual nilpotence
type: Item
instance of: scholarly article
title: On modular relation modules and residual nilpotence
zbMATH Open document ID: 0785.20006
DOI: 10.1016/0022-4049(93)90014-K
author: Andrew J. Gilchrist
author: Brian Hartley
published in: Journal of Pure and Applied Algebra
publication date: 1993-11-01T00:00:00Z
review text: Let \(F\) be a free non-cyclic group with a normal subgroup \(R\) and \(G=F/R\). Denote by \(S\) the \(n\)th dimension subgroup \(D_ n(R,p^ m)\), where \(p\) is a prime and \(n\), \(m\) are some integers. The following are equivalent: (i) \(F/S\) is residually nilpotent; (ii) \(F/S\) is residually a nilpotent \(p\)-group of finite exponent; (iii) \(G\) is residually a nilpotent \(p\)-group of finite exponent; (iv) the intersection of the powers of the augmentation ideal of \(G\) over \(\mathbb{F}_ p\) is zero. Suppose now that \(M\) is a verbal subgroup in \(R\) corresponding to a set of outer commutator words, \(M\subseteq \gamma_ c(R)\), where \(\gamma_ c(R)\) is the \(c\)th member of the lower central series in \(R\). Put \(\overline {S}=M/[M\cap \gamma_{c+1}(R)\gamma_ c(R)^ p]\). Suppose that \(\overline{S}\neq 0\) and \(G=F/R\) is infinite. Then \(\overline {S}\) is a faithful \(\mathbb{F}_ p G\)-module.
reviewed by: Vyacheslav A. Artamonov
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 427659
zbMATH Keywords: free non-cyclic group
zbMATH Keywords: \(n\)th dimension subgroup
zbMATH Keywords: residually nilpotent
zbMATH Keywords: augmentation ideal
zbMATH Keywords: verbal subgroup
zbMATH Keywords: commutator words
zbMATH Keywords: lower central series
MaRDI profile type: Publication
cites work: Q5340151
cites work: The residual nilpotence of certain presentations of finite groups
cites work: Q3658197
cites work: On the higher relation modules of a finite group
cites work: The Residual Nilpotence of Wreath Products
cites work: A note on free presentations and residual nilpotence
cites work: Q3218301
cites work: Relative higher relation modules, and a property of irreducible K- representations of \(GL_ n(K)\)
cites work: Q3815471
cites work: Homology of Higher Relation Modules and Torsion in Free Central Extensions of Groups
cites work: On minimal relation modules and 1-cohomology of finite groups
cites work: Sur les groupes nilpotents et les anneaux de Lie
cites work: The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups
cites work: Q4145882
cites work: Varieties of groups
cites work: Free presentations and relation modules of finite groups
cites work: Trace ideals of relation modules of finite groups
description: scientific article; zbMATH DE number 427659