Magnus embeddings and residual nilpotence
type: Item
instance of: scholarly article
title: Magnus embeddings and residual nilpotence
zbMATH Open document ID: 0622.20002
DOI: 10.1016/0021-8693(87)90023-8
author: Chander K. Gupta
author: Inder Bir S. Passi
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let G be a group, \({\mathbb{Z}}G\) its integral group ring, and \(\Delta\) (G) the augmentation ideal. Put \(\Delta^{\omega}(G)=\cap^{\infty}_{m=1}\Delta^ m(G)\). If \(G=F/R\), where F is a non cyclic free group, then \(\Delta^{\omega}(G)=0\) if and only if \(\cap^{\infty}_{m=1}({\mathfrak f}^ m+{\mathfrak r})={\mathfrak r}\), where \({\mathfrak f}=\Delta (F)\) and \({\mathfrak r}=\Delta (R){\mathbb{Z}}F\). For \(q\in \{0,1\}\), write \({\mathfrak r}^ q={\mathfrak r}\) if \(q=1\) and \({\mathfrak r}^ q={\mathfrak f}\) if \(q=0\). Then a theorem of \textit{C. K. Gupta} and \textit{N. D. Gupta} [Houston J. Math. 7, 43-52 (1981; Zbl 0467.20010)] states that \[ \cap^{\infty}_{m=1}({\mathfrak f}^ m+{\mathfrak r}^{q_ 1}...{\mathfrak r}^{q_ n})={\mathfrak r}^{q_ 1}...{\mathfrak r}^{q_ n}\text{ if and only if }\cap^{\infty}_{m=1} ({\mathfrak f}^ m+{\mathfrak r})={\mathfrak r}, \] that is, the augmentation ideal of G is residually nilpotent. Here \((q_ 1,...,q_ n)\) is an n-tuple different from (0,0,...,0) with \(q_ i\in \{0,1\}\). Write \(M(q_ 1,...,q_ n)=F\cap (1+{\mathfrak r}^{q_ 1}...{\mathfrak r}^{q_ n})\). In the same paper it is further claimed that \(\Delta^{\omega}(G)=0\) if and only if \(\Delta^{\omega}(F/M(q_ 1,...,q_ n))=0\). In the paper under review it is pointed out that the proof is incorrect, and the authors are able to go some distance towards re-proving this assertion. Firstly, if \(\Delta^{\omega}(F/R)=0\), then \(\Delta^{\omega}(F/M(q_ 1,...,q_ n))=0\). This is proved by using a faithful representation of \(F/M(q_ 1,...,q_ n)\), due to Gupta and Gupta, by triangular matrices whose entries are polynomials with coefficients in the group ring of a direct power of \(G=F/R\). In the converse direction the important case is when \(M(q_ 1,...,q_ n)\) is minimally expressed, that is, it differs from \(M(q_ 1,...,q_{n-1})\) and \(M(q_ 2,...,q_ n)\). In that case, if G is periodic or torsion free, \((q_ 1,q_ n)\neq (0,0)\), and \(F/M(q_ 1,...,q_ n)\) is residuall…
reviewed by: Brian Hartley
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 4008633
zbMATH Keywords: integral group ring
zbMATH Keywords: augmentation ideal
zbMATH Keywords: free group
zbMATH Keywords: residually nilpotent
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90023-8
OpenAlex ID: W2008653391
cites work: Q3919870
cites work: Some properties of Fox subgroups of free groups
cites work: A note on free presentations and residual nilpotence
cites work: The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups
cites work: Q4406395
cites work: Group algebras
description: scientific article; zbMATH DE number 4008633