On the \(\mathcal X\Phi\)-hypercentre of finite groups.
type: Item
instance of: scholarly article
title: On the \(\mathcal X\Phi\)-hypercentre of finite groups.
zbMATH Open document ID: 1184.20019
author: L. A. Shemetkov
author: A. N. Skiba
published in: Journal of Algebra
publication date: 2009-10-13T00:00:00Z
review text: Let \(G\) be a finite group. A subgroup \(H\) of a group \(G\) is said to be \(s\)-permutable in \(G\) if \(HP=PH\) for all Sylow subgroups \(P\) of \(G\). \textit{A. N. Skiba} [J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)] defined \(H_{sG}\) to be the subgroup of \(H\) generated by all those subgroups of \(H\) which are \(s\)-permutable in \(G\) and called \(H\) weakly \(s\)-permutable in \(G\) if \(G\) has a subnormal subgroup \(T\) such that \(HT=G\) and \(T\cap H\) is a subgroup of \(H_{sG}\). Let \(\mathcal U\) be the class of all finite supersoluble groups. A chief factor \(H/K\) is said to be \(\mathcal U\)-central if the semidirect product of \(H/K\) by \(G/C_G(H/K)\) is supersoluble, and the product of all normal subgroups \(H\) of \(G\) such that all non-Frattini \(G\)-chief factors of \(H\) are \(\mathcal U\)-central is denoted by \(Z_{\mathcal U\Phi}(G)\). The main result (Theorem 1.4) of the present paper is as follows. Let \(X\leq E\) be normal subgroups of \(G\). Suppose that every non-cyclic Sylow subgroup \(P\) of \(X\) has a subgroup \(D\) such that \(1<|D|<|P|\) and every subgroup \(H\) of \(P\) with \(|H|=|D|\) and every cyclic subgroup of \(P\) of order 4 (if \(|D|=2\) and \(P\) is a non-Abelian 2-group) is weakly \(s\)-permutable in \(G\). If \(X\) is either \(E\) or \(F^*(E)\), the generalised Fitting subgroup of \(E\), then \(E\leq Z_{\mathcal U\Phi}(G)\).
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5614702
zbMATH Keywords: \({\mathcal X}\Phi\)-hypercentre
zbMATH Keywords: weak permutability
zbMATH Keywords: weakly \(s\)-permutable subgroups
zbMATH Keywords: Sylow subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.03.029
OpenAlex ID: W2038240990
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DOI: 10.1016/J.JALGEBRA.2009.03.029
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description: scientific article; zbMATH DE number 5614702