A question of R. Maier concerning formations
type: Item
instance of: scholarly article
title: A question of R. Maier concerning formations
zbMATH Open document ID: 0855.20018
author: Adolfo Ballester-Bolinches
author: M. D. Pérez-Ramos
published in: Journal of Algebra
publication date: 1997-01-29T00:00:00Z
review text: The paper answers affirmatively the following question of \textit{R. Maier}: ``Let \(\mathfrak f\) be a saturated formation. If \(G=HK\) with \(\mathfrak f\)-groups \(H\) and \(K\) such that every subgroup of \(H\) is permutable with every subgroup of \(K\), then \(G\) belongs to \(\mathfrak f\) provided that \(\mathfrak f\) contains the saturated formation of all supersoluble groups. Does the above result extend to non-saturated formations which contain all supersoluble groups? The main result of this paper is: Let \(\mathfrak f\) be a formation containing the class of all supersoluble groups. Suppose that the group \(G=HK\) is the product of the totally permutable subgroups \(H\) and \(K\) (i.e. every subgroup of \(H\) is permutable with every subgroup of \(K\)). If \(H\) and \(K\) belong to \(\mathfrak f\), then \(G\) belongs to \(\mathfrak f\).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 922330
zbMATH Keywords: saturated formations
zbMATH Keywords: supersoluble groups
zbMATH Keywords: permutable subgroups
reviewed by: Rodica Covaci
MaRDI profile type: Publication
full work available at URL: jabr.1996.0198
OpenAlex ID: W1980210047
DOI: 10.1006/JABR.1996.0198
description: scientific article; zbMATH DE number 922330