Some properties of groups which allow homomorphisms onto their direct square
type: Item
instance of: scholarly article
title: Some properties of groups which allow homomorphisms onto their direct square
zbMATH Open document ID: 0822.20034
author: Ron Hirshon
published in: Journal of Algebra
publication date: 1994-08-16T00:00:00Z
review text: The author calls a nontrivial group \(G\) an M.B. group if \(G \times G\) is a homomorphic image of \(G\). The first construction of a finitely presented M.B. group was obtained by Baumslag and Miller in 1988. A general method generalizing the previous construction was given by Hirshon and Meier in 1992. The aim of this paper is to gather some properties of M.B. groups. For example: Corollary 2. A necessary and sufficient condition for a non-trivial group \(G\) to be an M.B. group is that there exists a homomorphic image \(A\) of \(G\) such that \(G\) is also a homomorphic image of \(A\) and \(A^ 2 \cong A\). Corollary 6. There exists a finitely generated group \(A\) such that \(A\) contains a finitely presented M.B. group \(G\) as a direct factor with \(G\) containing as a subgroup an isomorphic copy of any finitely presented group and such that \(A^ 2 \cong A\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 614208
zbMATH Keywords: Hopfian groups
zbMATH Keywords: direct square
zbMATH Keywords: homomorphic images
zbMATH Keywords: finitely presented M.B. groups
zbMATH Keywords: finitely generated groups
zbMATH Keywords: direct factor
MaRDI profile type: Publication
full work available at URL: jabr.1994.1185
OpenAlex ID: W2004536866
DOI: 10.1006/JABR.1994.1185
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description: scientific article; zbMATH DE number 614208