On join properties of Hall \(\pi\)-subgroups of finite \(\pi\)-soluble groups
type: Item
instance of: scholarly article
title: On join properties of Hall \(\pi\)-subgroups of finite \(\pi\)-soluble groups
zbMATH Open document ID: 0914.20021
published in: Journal of Algebra
publication date: 1999-06-17T00:00:00Z
review text: Let \(\pi\) be a set of primes and let \(\text{Hall}_\pi(G)\) denote the set of all Hall \(\pi\)-subgroups of a finite group \(G\). \(G\) has property \(J_\pi\) (resp. \(J_\pi^*\)) if, whenever \(H\in\text{Hall}_\pi(G)\) and \(U,V\leq G\) such that \(H\cap U\in\text{Hall}_\pi(U)\) and \(H\cap V\in\text{Hall}_\pi(V)\), then \(H\cap\langle U,V\rangle\in\text{Hall}_\pi(\langle U,V\rangle)\) (resp. \(\langle H\cap U,H\cap V\rangle\in\text{Hall}_\pi(\langle U,V\rangle)\)) holds. The authors among others show that these properties are equivalent, if \(G\) is \(\pi\)-soluble, to \(G/O_\pi(G)\) (resp. \(G\)) has a normal \(\pi\)-complement. Furthermore they investigate the following question: Let \(H\in\text{Hall}_\pi(G)\) and \(U,V\leq G\) be such that \(H\cap U\in\text{Hall}_\pi(U)\) and \(H\cap V\in\text{Hall}_\pi(V)\). Which conditions imply that \(\langle H\cap U,H\cap V\rangle\in\text{Hall}_\pi(\langle U,V\rangle)\)? In particular they derive an improvement of Wielandt's result on the behaviour of Hall \(\pi\)-subgroups in groups generated by subnormal subgroups.
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 1218673
zbMATH Keywords: Fitting formations
zbMATH Keywords: quasinilpotent groups
zbMATH Keywords: residuals
zbMATH Keywords: Hall \(\pi\)-subgroups
zbMATH Keywords: finite groups
zbMATH Keywords: normal \(\pi\)-complements
zbMATH Keywords: subnormal subgroups
author: Adolfo Ballester-Bolinches
author: Luis M. Ezquerro
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
full work available at URL: jabr.1997.7356
OpenAlex ID: W2014969722
cites work: Finite soluble groups
cites work: Finite groups in which all \(p\)-subnormal subgroups form a lattice
cites work: An intersection property of Sylow \(p\)-subgroups affecting \(p\)-length in finite \(p\)-solvable groups
cites work: Q3659962
cites work: Q3744347
cites work: Sylowgruppen und Kompositions-Struktur
DOI: 10.1006/JABR.1997.7356
description: scientific article; zbMATH DE number 1218673