On a conjecture of O. ~I. ~Tavgen'
type: Item
instance of: scholarly article
title: On a conjecture of O. ~I. ~Tavgen'
zbMATH Open document ID: 0941.20025
DOI: 10.1007/BF02674903
author: A. S. Shtern
author: A. N. Zubkov
published in: Siberian Mathematical Journal
publication date: 2000-04-13T00:00:00Z
review text: This article is devoted to a conjecture of \textit{O.~I.~Tavgen'} about a group-theoretic characterization of linearity for an amalgamated product of two groups [Dokl. Akad. Nauk, Ross. Akad. Nauk 328, No. 2, 153-156 (1993; Zbl 0830.20071)]. The authors give a negative answer to a more general conjecture of O.~Tavgen' about Artin-Rees properties: Let \(G\) be an analytic pro-\(p\)-group and let \(M\) be a finitely generated (in the topological sense) \(\mathbb{Z}_p\)-torsion free \(\mathbb{Z}_pG\)-module (i.e., \(M\) is free as an Abelian pro-\(p\)-group). Let \(U\) be a closed submodule and let \(\dim_{\mathbb{Z}_p}U<\infty\). Then there is a closed submodule \(N\subseteq M\) such that \(N\cap U=0\) and \(\dim_{\mathbb{Z}_p}M/N<\infty\).
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20H25
zbMATH DE Number: 1431904
zbMATH Keywords: amalgamated products of groups
zbMATH Keywords: linearity of groups
zbMATH Keywords: analytic pro-\(p\)-groups
Wikidata QID: Q123248757
MaRDI profile type: Publication
cites work: Groups of integral representation type
cites work: A group theoretic characterization of linear groups
cites work: \(p\)-adic analytic groups
cites work: Q4064209
cites work: Q5182382
cites work: Q3997907
full work available at URL: https://doi.org/10.1007/bf02674903
OpenAlex ID: W2073673804
description: scientific article; zbMATH DE number 1431904