Groups of operators that fix a Sylow subgroup
type: Item
instance of: scholarly article
title: Groups of operators that fix a Sylow subgroup
zbMATH Open document ID: 1028.20018
author: Enrico Jabara
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2003-10-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108556
review text: Let \(A\) and \(G\) be finite groups and suppose that \(A\) acts on \(G\). \(A\) is said to be an \(S\)-group of operators of \(G\) if for each prime number \(r\) such that \(r\mid|G|\), \(G\) has a unique \(A\)-invariant \(r\)-Sylow subgroup. The group \(G\) is said to be \(A\)-\(p\)-nilpotent if \(A\) is an \(S\)-group and \(|G|=p^n\), \(p\) a prime number. \textit{A. Beltrán} proved [in Arch. Math. 69, No. 3, 177-184 (1997; Zbl 0917.20005)] that if \((|G|,|A|)=1\) and \(C_G(A)\) is a nilpotent group, then \(G\) is a soluble group. The author proves that the \(A\)-\(p\)-nilpotent groups are \(p\)-nilpotent and soluble groups. He also obtains that the Fitting length of \(G\) is bounded by \(n+2\) and gives some counter-example to show that the bound cannot be improved.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 1993738
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: groups of operators
zbMATH Keywords: soluble groups
zbMATH Keywords: nilpotent groups
reviewed by: Francisco Pérez-Monasor
MaRDI profile type: Publication
cites work: Actions with nilpotent fixed point subgroup
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cites work: Q5577154
cites work: On the p -Length of p -Soluble Groups and Reduction Theorems for Burnside's Problem
cites work: The existence of regular orbits for nilpotent groups
cites work: On finite groups admitting a coprime automorphism of prime order
cites work: Fitting height of \(A\)-nilpotent groups.
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description: scientific article; zbMATH DE number 1993738