Groups with finite automorphism classes of subgroups
type: Item
instance of: scholarly article
title: Groups with finite automorphism classes of subgroups
zbMATH Open document ID: 0653.20035
author: Federico Menegao
author: Howard Smith
author: James Wiegold
author: John C. Lennox
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108110
review text: \textit{D. J. S. Robinson} and \textit{J. Wiegold} [ibid. 71, 273-286 (1984; Zbl 0543.20025)] had investigated groups in which the orbits of all subgroups under the automorphism group are boundedly finite, and had shown that this is the case if (and trivially only if) the orbits of only all abelian subgroups are boundedly finite; and had given another, more complicated equivalent condition. In the present paper the authors show that if in a group the orbits of all subgroups are finite, then they are boundedly finite. The proof is elementary, but (perhaps not surprisingly) quite hard. The authors then go on to study groups in which the orbits of subgroups under the endomorphism semigroup are finite; again this is equivalent to them being boundedly finite, and also to the same condition for the abelian subgroups only; and the structure of the group can be quite simply described. Finally they show that a group in which all elements have boundedly finite orbits under the endomorphism semigroup must be finite. Here ``boundedly'' can not be omitted. This generalizes a well-known theorem of \textit{R. Baer} [Trans. Am. Math. Soc. 79, 521-540 (1955; Zbl 0065.010)], which says that a group is finite if its endomorphism semigroup is finite.
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E18
zbMATH DE Number: 4065279
zbMATH Keywords: automorphism group
zbMATH Keywords: orbits of subgroups
zbMATH Keywords: endomorphism semigroup
zbMATH Keywords: abelian subgroups
zbMATH Keywords: boundedly finite orbits
reviewed by: B. H. Neumann
MaRDI profile type: Publication
cites work: Finite Extensions of Abelian Groups With Minimum Condition
cites work: Q3293433
cites work: Q5609457
cites work: Endliche Gruppen I
cites work: Groups with finite classes of conjugate subgroups
cites work: Groups with boundedly finite automorphism classes
cites work: Multiplicators and groups with finite central factor-groups
description: scientific article; zbMATH DE number 4065279