The Fitting length of the Hughes subgroup
type: Item
instance of: scholarly article
title: The Fitting length of the Hughes subgroup
zbMATH Open document ID: 0604.20021
DOI: 10.1016/0021-8693(87)90201-8
author: Alberto Espuelas
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The Fitting length of a solvable group is the least possible length of its normal series with nilpotent factors. Theorem: Let \(G\) be a finite solvable group having a normal subgroup \(H\neq G\) such that every element of \(G\setminus H\) is a \(p\)-element of order at most \(p^ n\), \(p\) odd. Then the Fitting length of \(H\) is at most \(n\). The bound is best possible. This theorem generalizes and strengthens earlier results of \textit{O. H. Kegel} [Math. Z. 75, 373-376 (1961; Zbl 0104.24904)] (the case \(n=1\)) and \textit{H. Kurzweil} [Arch. Math. 22, 136-145 (1971; Zbl 0218.20022)] who proved that Fitting length \(\leq 2n-2\) under the additional condition that \(p\) is not an exceptional prime in the Hall-Higman sense. The conditions of the theorem generalize the case of a fixed-point-free automorphism of order \(p^ n\) acting on a finite solvable \(p'\)-group. The Hall-Higman methods are used in the proof. The case \(p=2\) remains still open.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 3975194
zbMATH Keywords: Hughes subgroup
zbMATH Keywords: Hall-Higman theorem
zbMATH Keywords: Fitting length
zbMATH Keywords: normal series
zbMATH Keywords: nilpotent factors
zbMATH Keywords: finite solvable group
zbMATH Keywords: fixed-point-free automorphism
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90201-8
OpenAlex ID: W2057285571
cites work: Carter subgroups and Fitting heights of finite solvable groups
cites work: Q3934684
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cites work: On groups admitting fixed point free abelian operator groups
description: scientific article; zbMATH DE number 3975194