Outer automorphism groups of metabelian groups
type: Item
instance of: scholarly article
title: Outer automorphism groups of metabelian groups
zbMATH Open document ID: 0968.20020
DOI: 10.1016/S0022-4049(99)00022-5
author: Rüdiger Göbel
author: Agnes T. Paras
published in: Journal of Pure and Applied Algebra
publication date: 2000-11-13T00:00:00Z
review text: Given a group \(G\), the outer automorphism group \(\text{Out }G\) of \(G\) is given as \(\Aut G/\text{Inn }G\), the quotient of the group of all automorphisms by the group of inner automorphisms. There are a number of results about automorphisms of metablian groups which are reviewed here before the proof of the main result. This states that, for any group \(M\), there exists a torsion-free metabelian group \(G\) with trivial centre such that \(M\) is isomorphic to \(\text{Out }G\). The proof involves a long and detailed examination of the situation. In a note added in proof, the authors announce that they have improved the result to: If \(M\) is any group of cardinality \(\kappa<2^{\aleph_0}\) then there is a torsion-free metabelian group \(G\) of cardinality \(|G|=\max\{\kappa,\aleph_0\}\) with \(\text{Out }G\cong M\).
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 08A35
Mathematics Subject Classification ID: 20K20
zbMATH DE Number: 1472407
zbMATH Keywords: groups of automorphisms
zbMATH Keywords: outer automorphism groups
zbMATH Keywords: inner automorphisms
zbMATH Keywords: torsion-free metabelian groups
reviewed by: John D. P. Meldrum
MaRDI profile type: Publication
full work available at URL: s0022-4049(99)00022-5
OpenAlex ID: W2018058392
description: scientific article; zbMATH DE number 1472407