Products of groups and group classes
type: Item
instance of: scholarly article
title: Products of groups and group classes
zbMATH Open document ID: 0809.20017
DOI: 10.1007/BF02772979
author: Andrew Fransman
author: Bernhard Amberg
published in: Israel Journal of Mathematics
publication date: 1994-09-27T00:00:00Z
review text: This paper mainly concerns finite triply factorized groups \(G = AB = AC = BC\), where \(A\), \(B\) and \(C\) are subgroups. Generalizing previous results of \textit{O. H. Kegel} [Math. Z. 87, 42-48 (1965; Zbl 0123.02503)] and Peterson (unpublished) the authors prove the following theorem: Let \(\mathfrak X\) be a Schunck class of finite groups and let the finite group \(G = AB = AC = BC\) be the product of two nilpotent subgroups \(A\) and \(B\) and an \(\mathfrak X\)-subgroup \(C\). If for every common prime divisor \(p\) of the orders of \(A\) and \(B\) the group of order \(p\) belongs to \(\mathfrak X\), then \(G\) is a (soluble) \(\mathfrak X\)-group. This theorem becomes false if only one of the subgroups \(A\) and \(B\) is nilpotent (and the other is an \(\mathfrak X\)-group). However the authors prove that if \(\mathfrak X\) is a Schunck class of finite soluble groups (or a saturated formation of finite groups) and the finite group \(G = AB = AK = BK\) is the product of two \(\mathfrak X\)-subgroups \(A\) and \(B\) and a nilpotent normal subgroup \(K\), then \(G\) is an \(\mathfrak X\)-group. An example is exhibited to show that the latter result is no longer true for arbitrary Schunck classes. Finally, in the last section this result is extended to some particular classes of infinite groups.
reviewed by: Silvana Franciosi
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F17
zbMATH DE Number: 645141
zbMATH Keywords: products of nilpotent subgroups
zbMATH Keywords: finite triply factorized groups
zbMATH Keywords: finite soluble groups
zbMATH Keywords: saturated formations of finite groups
zbMATH Keywords: Schunck classes
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02772979
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description: scientific article; zbMATH DE number 645141