Elementary abelian operator groups
type: Item
instance of: scholarly article
title: Elementary abelian operator groups
zbMATH Open document ID: 0799.20021
DOI: 10.1007/BF02567690
published in: Manuscripta Mathematica
publication date: 1994-11-08T00:00:00Z
full work available at URL: https://eudml.org/doc/155920
review text: Let \(A\) be an elementary abelian \(p\)-group which acts on a solvable \(p'\)- group \(G\). If \(\phi \in A\), \(C_ G(\phi)\) denotes the fixed point subgroup of \(\phi\). Previous results of \textit{A. Turull} [J. Algebra 86, 555-566 (1984; Zbl 0526.20017)] and of \textit{F. Gross} [Bull. Aust. Math. Soc. 7, 91-100 (1972; Zbl 0241.20016)] indicate that there is a strong connection between the nilpotency length \(l(G)\) of \(G\) and the nilpotency length of the fixed point subgroups of the nontrivial elements of \(A\). The paper contains two main results: 1) Let \(n\) be a positive integer and let \(G\) be a finite solvable \(p'\)-group acted upon by an elementary abelian \(p\)-group \(A\) of order \(p^{n + 1}\). If \(l(C_ G(\phi)) \leq n\) for every nontrivial element \(\phi\) of \(A\), then \(l(G) \leq n + 1\). This theorem shows that the results of Turull and Gross (loc. cit.) provide the best bounds for \(l(G)\) in their cases. 2) Let \(G\) be a periodic solvable \(2'\)-group acted upon by an elementary abelian group \(A\) of order 8. Suppose that \(C_ G(\phi)\) has a nilpotent derived subgroup for every nontrivial element \(\phi\) of \(A\). Then \(l(G') \leq 2\).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 554381
zbMATH Keywords: finite solvable groups
zbMATH Keywords: Fitting length
zbMATH Keywords: elementary abelian \(p\)-group
zbMATH Keywords: fixed point subgroup
zbMATH Keywords: nilpotency length
zbMATH Keywords: periodic solvable \(2'\)-group
zbMATH Keywords: nilpotent derived subgroup
author: Pavel Shumyatsky
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
cites work: Q5577154
cites work: Elementary abelian operator groups
cites work: Some sufficient conditions for a group to be nilpotent
cites work: A four group of automorphisms with a small number of fixed points
cites work: On periodic soluble groups and the fixed point groups of operators
cites work: On locally soluble periodic groups with Chernikov centralizer of a four-subgroup
cites work: On periodic solvable groups having automorphisms with nilpotent fixed point groups
cites work: Fitting height of groups and of fixed points
cites work: Automorphisms of Finite Groups and their Fixed-Point Groups
OpenAlex ID: W2030819221
description: scientific article; zbMATH DE number 554381