A note on groups with countable automorphism groups
type: Item
instance of: scholarly article
title: A note on groups with countable automorphism groups
zbMATH Open document ID: 0582.20024
DOI: 10.1007/BF01202494
author: Silvana Franciosi
author: F. de Giovanni
published in: Archiv der Mathematik
publication date: 1986-01-01T00:00:00Z
review text: The subject of this paper is the analysis of the behaviour of the elements of finite order of a group G when countability conditions are imposed on the automorphism group Aut G of G. The main result is the following Theorem. Let G be a finite-by-nilpotent group such that the Hirsch-Plotkin radical \((=\) the maximum normal locally nilpotent subgroup) of Aut G is countable; then the Sylow subgroups of G have finite exponents, and G has finite exponent, if it is a torsion group. A stronger result is obtained when the group of all central automorphisms of G is countable. Examples show that these results are not true if the countability condition is imposed on the Baer radical of Aut G.
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 3931256
zbMATH Keywords: elements of finite order
zbMATH Keywords: countability conditions
zbMATH Keywords: automorphism group
zbMATH Keywords: finite-by-nilpotent group
zbMATH Keywords: Hirsch-Plotkin radical
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: central automorphisms
MaRDI profile type: Publication
cites work: Finite Extensions of Abelian Groups With Minimum Condition
cites work: On the number of outer automorphisms of an infinite nilpotent p-group
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cites work: Q5183744
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cites work: Homology in group theory
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full work available at URL: https://doi.org/10.1007/bf01202494
OpenAlex ID: W2055488782
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description: scientific article; zbMATH DE number 3931256