On the centralizer and the commutator subgroup of an automorphism.
type: Item
instance of: scholarly article
title: On the centralizer and the commutator subgroup of an automorphism.
zbMATH Open document ID: 1270.20031
author: Gérard Endimioni
author: Primož Moravec
published in: Monatshefte für Mathematik
publication date: 2012-11-21T00:00:00Z
review text: Let \(\varphi\) be an automorphism of a group \(G\). It is proved that if \(G\) is polycyclic, \(|\varphi|=2\), and \(C_G(\varphi )\) is finite, then \([G,\varphi]'\) is finite. It is also proved that if \(G\) is polycyclic and \(C_G(\varphi)\) is finite, then \(G/[G,\varphi]\) is finite (for any \(\varphi\)). If in addition \(|\varphi|\) is finite, then \(|G:C_G(\varphi)[G,\varphi]|\) is finite. There are also results on interrelation between \(C_G(\varphi)\) and \([G,\varphi]\) for metabelian groups with automorphisms of finite order, and for soluble groups with an automorphism of order 2.
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 6109325
zbMATH Keywords: automorphisms of finite order
zbMATH Keywords: polycyclic groups
zbMATH Keywords: metabelian groups
zbMATH Keywords: soluble groups
zbMATH Keywords: finiteness of centralizers
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
full work available at URL: s00605-011-0298-0
OpenAlex ID: W1980094046
cites work: Q3218303
cites work: On almost regular automorphisms.
cites work: Polycyclic group admitting an almost regular automorphism of prime order.
cites work: Groups and Rings Having Automorphisms without Non-Trivial Fixed Elements
cites work: Nilpotency of solvable groups admitting a splitting automorphism of prime order
cites work: Q4820338
cites work: Q5441287
DOI: 10.1007/S00605-011-0298-0
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description: scientific article; zbMATH DE number 6109325