On the Fitting series of the product of two finite nilpotent groups
type: Item
instance of: scholarly article
title: On the Fitting series of the product of two finite nilpotent groups
zbMATH Open document ID: 0811.20021
author: Gemma Parmeggiani
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1995-04-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108324
review text: Let the finite group \(G = AB\) be the product of two nilpotent subgroups \(A\) and \(B\). Then \(G\) is soluble by a well-known theorem of Kegel and Wielandt, and the structure of such a group has been the object of many investigations. In particular, it has recently been proved by \textit{S. Franciosi, H. Heineken, M. L. Newell} and the reviewer [Arch. Math. 57, 313-318 (1991; Zbl 0774.20021)] that, if the subgroup \(A\) is abelian and \(F\) denotes the Fitting subgroup of \(G\), then the product \(AF\) is a normal subgroup of \(G\). The theorem proved in this article extends the above result to an arbitrary finite product \(G = AB\) of two nilpotent subgroups. In fact, if \(F_ i(G)\) denotes the \(i\)-th term of the Fitting series of \(G\) (\(i \geq 1\)) and \(d\) is the derived length of \(A\), it is shown that \(AF_{2n - 1} (G)\) is a normal subgroup of \(G\) for every \(n \geq d\). In particular, \(G\) has Fitting length at most \(2d + 1\), and an example proves that this bound is the best possible.
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 627435
zbMATH Keywords: factorized groups
zbMATH Keywords: product of nilpotent subgroups
zbMATH Keywords: finite groups
zbMATH Keywords: Fitting series
zbMATH Keywords: derived length
zbMATH Keywords: Fitting length
zbMATH Keywords: Fitting subgroup
zbMATH Keywords: normal subgroup
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 627435