Groups with trivial virtual automorphism group
type: Item
instance of: scholarly article
title: Groups with trivial virtual automorphism group
zbMATH Open document ID: 0722.20022
DOI: 10.1007/BF02773748
author: F. Menegazzo
author: M. J. Tomkinson
published in: Israel Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The authors consider groups G with the following property: Let H,K be subgroups of finite index in G and \(\phi: H\to K\) an isomorphism. Then there exists a subgroup L of \(H\cap K\) of finite index in G such that \(\phi\) is the identity on L. A finite group has this property. Mann and Lubotzky asked whether there exist infinite groups with this property. The authors answer this question by giving two examples of infinite groups having this property. The first example is an extraspecial 2-group of cardinality \(2^{\aleph_ 0}\) and the second example is a 2-group again of cardinality \(2^{\aleph_ 0}\) with derived subgroup isomorphic to a 4-group. It is an open problem whether there exists a group of countable cardinality with this property. They use a result of \textit{F. Menegazzo} and \textit{D. J. S. Robinson} [Rend. Semin. Mat. Univ. Padova 78, 267-277 (1987; Zbl 0637.20017)] and prove first that such a group G is virtually a 2-group with finite central derived subgroup and \(G/G''\) elementary abelian.
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 4190019
zbMATH Keywords: virtual automorphism group
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: infinite groups
zbMATH Keywords: extraspecial 2-group
zbMATH Keywords: finite central derived subgroup
MaRDI profile type: Publication
cites work: Q5670800
cites work: A finiteness condition on automorphism groups
cites work: Extraspecial p-groups
cites work: Q4078224
full work available at URL: https://doi.org/10.1007/bf02773748
OpenAlex ID: W2030948697
description: scientific article; zbMATH DE number 4190019