On torsion in metabelian pro-p-groups with one defining relation
type: Item
instance of: scholarly article
title: On torsion in metabelian pro-p-groups with one defining relation
zbMATH Open document ID: 0546.20022
DOI: 10.1007/BF01137693
author: A. G. Pshenichnyj
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: Theorem. Let F be the free metabelian pro-p-group freely generated by \(x_ 1,...,x_ n\), let X be the normal closure of \(x_ 1\), and let R be generated by \(r=x_ 1^{p^{\ell}}c\), where \(c\in F'\) and \(\ell\geq 1\). Let \(\alpha:{\mathbb{Z}}_ p\to {\mathbb{Z}}/p{\mathbb{Z}}\) be the natural homomorphism and \(d_ i\) the partial derivative with respect to \(x_ i\) of \({\mathbb{Z}}_ pF\) in Z'\({}_ pF/F'X\). Then F/R is torsion-free if and only if there is an i such that \(\alpha d_ i(r)\neq 0\).
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 3869585
zbMATH Keywords: torsion-freeness condition
zbMATH Keywords: free metabelian pro-p-group
zbMATH Keywords: partial derivative
MaRDI profile type: Publication
cites work: Q3208874
full work available at URL: https://doi.org/10.1007/bf01137693
OpenAlex ID: W2046859611
description: scientific article; zbMATH DE number 3869585