A finiteness condition on automorphism groups
type: Item
instance of: scholarly article
title: A finiteness condition on automorphism groups
zbMATH Open document ID: 0637.20017
author: F. Menegazzo
author: Derek J. S. Robinson
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108082
review text: The automorphism \(\alpha\) of the group \({\mathcal G}\) is called ``virtually trivial'' if its group of fixed elements has finite index in \({\mathcal G}\). (Such automorphisms have also been called ``bounded'' automorphisms.) Groups with only virtually trivial automorphisms are here called VTA- groups. They are FC-groups (that is, groups with finite classes of conjugate elements), and are shown to be even FD-groups (that is, groups with finite derived groups). Moreover, the automorphism group of a VTA- group is locally finite; moreover the centre of the group is reduced and its primary components are all finite. If a VTA-group is periodic, then it has finite exponent and its centre is finite. Criteria for a group to be a VTA-group are given, as well as a number of interesting examples.
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 4038079
zbMATH Keywords: virtually trivial automorphisms
zbMATH Keywords: VTA-groups
zbMATH Keywords: FC-groups
zbMATH Keywords: FD-groups
zbMATH Keywords: finite derived groups
zbMATH Keywords: automorphism group
reviewed by: B. H. Neumann
MaRDI profile type: Publication
cites work: Groups Covered By Permutable Subsets
cites work: Q5654235
cites work: A Contribution to the Theory of Groups with Finitely Many Automorphisms
cites work: Q3961021
cites work: Groups with boundedly finite automorphism classes
cites work: Locally soluble FC-groups
description: scientific article; zbMATH DE number 4038079