On normally embedded projectors in finite soluble groups
type: Item
instance of: scholarly article
title: On normally embedded projectors in finite soluble groups
zbMATH Open document ID: 0837.20024
author: Adolfo Ballester-Bolinches
author: Luis M. Ezquerro
published in: Journal of Algebra
publication date: 1996-01-14T00:00:00Z
review text: Let \(NE(G)\) be the set of all normally embedded subgroups of a group \(G\). The authors start from the question: what saturated formations \(\mathcal F\) of characteristic \(\pi\) satisfy the following property: (a) In each group \(G\), the set of primes dividing the order of the \(\mathcal F\)-projectors of \(G\) coincides with the set of primes dividing the order of the Hall \(\pi\)-subgroups of \(G\). This problem is closely related to the question of characterizing the saturated formations \(\mathcal F\) for which the class \(N_{\mathcal F}\), of all soluble groups \(G\) such that every \(\mathcal F\)-projector of \(G\) is in \(NE(G)\), is a saturated formation. The answer to the above mentioned problems is given by proving that the saturated formations \(\mathcal F\) satisfying the property (a) are exactly those for which \(N_{\mathcal F}\) is a saturated formation. Further, the authors characterize those saturated formations \(\mathcal F\) for which the formation \(S_{\mathcal F}\), of all soluble groups \(G\) such that any \(\mathcal F\)-normalizer of \(G\) is in \(NE(G)\), is saturated. A useful description of the canonical local definition of the saturation of \(S_{\mathcal F}\) and a study of the saturation of the formation \(\mathcal X\) of the \(A\)-groups are also given.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 823519
zbMATH Keywords: order of \(\mathcal F\)-projectors
zbMATH Keywords: normally embedded subgroups
zbMATH Keywords: saturated formations
zbMATH Keywords: Hall \(\pi\)-subgroups
zbMATH Keywords: soluble groups
zbMATH Keywords: local definition
zbMATH Keywords: saturation
reviewed by: Rodica Covaci
MaRDI profile type: Publication
full work available at URL: jabr.1995.1313
OpenAlex ID: W2080568343
DOI: 10.1006/JABR.1995.1313
description: scientific article; zbMATH DE number 823519