Finite soluble groups with supersoluble Sylow normalizers
type: Item
instance of: scholarly article
title: Finite soluble groups with supersoluble Sylow normalizers
zbMATH Open document ID: 0638.20013
DOI: 10.1007/BF01313488
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let G be a finite group. It was proved by \textit{M. Bianchi}, \textit{A. Gillio Berta Mauri} and \textit{P. Hauck} [Arch. Math. 47, 193-197 (1986; Zbl 0605.20017)] that if every normalizer of Sylow p-subgroups of G (p\(\in \pi (G))\) is nilpotent, then G is nilpotent. The corresponding result is false if one relaxes the condition of nilpotency to that of supersolvability, since there are simple nonabelian groups having supersolvable Sylow normalizers. The authors consider the class \(N^{{\mathcal S}}\) of the finite groups all of whose Sylow normalizers are supersolvable and restrict to the solvable members of \(N^{{\mathcal L}}\). If \(G\in N^{{\mathcal S}}\) and \(\pi (G)=\{p,q\}\), \(p| q-1\), they prove that there exists \(H\in N^{{\mathcal S}}\), with \(\pi (H)=\pi (G)\) and \(X\leq H\) with \(X\cong G\). If G is soluble, \(G\in N^{{\mathcal S}}\) and \(| \pi (G)| \geq 3\), then it is not always possible to embed G in this manner in a solvable group \(H\in N^{{\mathcal S}}\) with \(\pi (H)=\pi (G)\). The following bound for the Fitting length h(G) is obtained for a solvable group \(G\in N^{{\mathcal S}}\); if \(\pi (G)=\{p_ 1,p_ 2,...,p_ n\}\), if \(l_ i(G)\leq l_ j(G)\) for \(i<j\), where \(l_ i(G)\) is the \(p_ i\)-length of G, then \(h(G)\leq l_ 1(G)+l_ 2(G)\). The smallest member p of \(\pi\) (G) has a special property if \(G\in N^{{\mathcal S}}\) is solvable and \(| \pi (G)| \geq 3:\) \(l_ p(G)\leq 2\).
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 4040162
zbMATH Keywords: p-length
zbMATH Keywords: supersolvable Sylow normalizers
zbMATH Keywords: solvable group
zbMATH Keywords: Fitting length
author: Valeria Fedri
author: Luigi Serena
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
cites work: On finite groups with nilpotent Sylow-normalizers
cites work: Two Applications of Twisted Wreath Products to Finite Soluble Groups
cites work: Q3659962
cites work: Endliche Gruppen I
cites work: Q5577154
cites work: On the p -Length of p -Soluble Groups and Reduction Theorems for Burnside's Problem
cites work: Connection between the 2-length and the derived length of a Sylow 2-subgroup of a finite solvable group
full work available at URL: https://doi.org/10.1007/bf01313488
OpenAlex ID: W2077633695
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description: scientific article; zbMATH DE number 4040162