A note on free presentations and residual nilpotence
type: Item
instance of: scholarly article
title: A note on free presentations and residual nilpotence
zbMATH Open document ID: 0538.20010
DOI: 10.1016/0022-4049(84)90023-9
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: A group H is said to be discriminated by nilpotent groups of prime power exponent if, given any finite set \(h_ 1,h_ 2,...,h_ n\) of non- trivial elements of H, there exists a nilpotent group K of (finite) prime power exponent and a homomorphism \(\phi:H\to K\) such that \(\phi(h_ i)\neq 1\) for \(i=1,2,...,n.\) Let \({\mathfrak D}\) denote the class of all groups that are either residually torsion-free nilpotent or discriminated by nilpotent groups of finite prime power exponent. The main result of this paper is the following Theorem. Let F be a non-cyclic free group, \(R\triangleleft F\), and S be a fully invariant subgroup of R such that R/S is non-trivial, torsion-free and nilpotent. Then the following conditions are equivalent: (a) F/S is residually nilpotent, (b) \(F/S\in {\mathfrak D},\) (c) \(F/R\in {\mathfrak D}.\)
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 3855364
zbMATH Keywords: residually nilpotent groups
zbMATH Keywords: free groups
zbMATH Keywords: discriminated by nilpotent groups
author: Brian Hartley
reviewed by: Inder Bir S. Passi
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90023-9
OpenAlex ID: W1977607090
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description: scientific article; zbMATH DE number 3855364