Extensions of residually finite groups
type: Item
instance of: scholarly article
title: Extensions of residually finite groups
zbMATH Open document ID: 0804.20023
author: P. R. Hewitt
published in: Journal of Algebra
publication date: 1995-01-17T00:00:00Z
review text: The author investigates the problem of the residual finiteness of extensions of residually finite groups. The following theorem is proved: ``Let \(G=\text{SL}_ d(\mathbb{Z})\), and \(M =\mathbb{Z}^ d\) its natural module. (1) If \(d=2\), then every extension of \(G\) over \(M\) -- or any other residually finite \(G\)-group -- is residually finite. (2) If \(d=3\), then \(H^ 2(G,M) =\mathbb{Z}/2^ t\oplus\mathbb{Z}\), for some positive integer \(t\). Moreover, a 2-class determines a residually finite extension if and only if it is a torsion class. (3) If \(d > 3\), but \(d\neq 5\), then \(H^ 2(G,M)=0\).'' From the other results we mention only one. An extension \(E\) of \(G\) over \(M\) is said to be virtually split if there is a subgroup of finite index in \(E\) that contains and is split over \(M\). For a residually finite \(G\)-group \(M\) and an extension \(E\) of \(G\) over \(M\) the author proves that \(E\) is residually finite if and only if \(E\) is residually virtually split.
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 554146
zbMATH Keywords: residual finiteness
zbMATH Keywords: extensions of residually finite groups
zbMATH Keywords: residually finite extensions
zbMATH Keywords: virtually split extensions
reviewed by: Alexander Ivanovich Budkin
MaRDI profile type: Publication
full work available at URL: jabr.1994.1041
OpenAlex ID: W2062938475
DOI: 10.1006/JABR.1994.1041
description: scientific article; zbMATH DE number 554146