On virtual properties and group extensions
type: Item
instance of: scholarly article
title: On virtual properties and group extensions
zbMATH Open document ID: 0358.20048
DOI: 10.1007/BF01214488
author: Hans Rudolf Schneebeli
published in: Mathematische Zeitschrift
publication date: 1978-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/172652
review text: Let \(C\) be a class of groups which is closed with respect to isomorphisms and subgroups of finite Index, and containing the trivial group 1. A group which contains a \(C\)-group of finite Index is called a virtual \(C\)-group. We investigate whether the class \(\mathrm{v}C\) of all virtual \(C\)-groups is extension closed. We construct examples showing that the following classes are not closed with respect to group extensions nor amalgamated free products: the virtual duality groups, the classes \(\mathrm{v}FP\), \(FH\), the groups of virtually finite cohomology dimension, the virtually torsion-free groups. A closure criterion is derived in order to establish results like: Let \(C\) denote one of the following classes: The surface groups, the finitely generated groups \(G\) with non-trivial centre and c\(\mathrm{cd}\;G\leq2\), all subgroups of the rationals. Then the virtual poly-\(C\)-groups are extension closed. Let \(C\) be the class of all solvable groups of finite cohomology dimension or the class of all solvable duality groups, then the class \(\mathrm{v}C\) is extension closed.
reviewed by: Hans Rudolf Schneebeli
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 3558086
MaRDI profile type: Publication
Wikidata QID: Q57308915
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description: scientific article; zbMATH DE number 3558086