On \(\mathcal F\)-supplemented subgroups of finite groups.
type: Item
instance of: scholarly article
title: On \(\mathcal F\)-supplemented subgroups of finite groups.
zbMATH Open document ID: 1172.20019
published in: Manuscripta Mathematica
publication date: 2008-12-02T00:00:00Z
review text: All groups involved in this review are finite. Recall that a formation is a class of groups which is closed under taking epimorphic images and subdirect products. If \(\mathcal F\) is a formation and \(G\) is a group, a subgroup \(H\) of \(G\) is said to be \(\mathcal F\)-supplemented in \(G\) if there exists a subgroup \(T\) of \(G\) such that \(G=TH\) and \((H\cap T)H_G/H_G\) is contained in the \(\mathcal F\)-hypercenter \(Z^{\mathcal F}_\infty(G/H_G)\) of \(G/H_G\), that is, the largest normal subgroup of \(G/H_G\) for which every chief factor of \(G/H_G\) below it is \(\mathcal F\)-central in \(G/H_G\). Of course normal or complemented subgroups (more generally, c-supplemented subgroups introduced by \textit{X. Guo, Y. Wang} and the reviewer [in Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)]) are examples of \(\mathcal F\)-supplemented subgroups for every non-empty formation \(\mathcal F\). In this paper, the influence of \(\mathcal F\)-supplemented subgroups on the structure of a group is analyzed. For instance, a group \(G\) is soluble if and only if every subgroup of prime order (respectively, every Sylow subgroup) is supplemented in \(G\) with respect to the formation of all soluble groups (Theorems 4.1 and 4.2). The author also proves that if \(\mathcal F\) is a subgroup-closed saturated formation containing all supersoluble groups and \(G\) is a group with a subgroup \(E\) containing the \(\mathcal F\)-residual of \(G\), then \(G\) belongs to \(\mathcal F\) provided that every subgroup of prime order or order \(4\) of \(E\) is supplemented in \(G\) with respect to the formation of all supersoluble groups. The same conclusion follows if the subgroups of prime order or order \(4\) are replaced by the family of all maximal subgroups of the non-cyclic Sylow subgroups of \(E\) not having a supersoluble supplement in \(G\). A series of known results about normal, c-normal and c-supplemented subgroups are unified and generalized. Remarks: It should b…
reviewed by: Adolfo Ballester-Bolinches
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 5377117
zbMATH Keywords: finite soluble groups
zbMATH Keywords: supersoluble groups
zbMATH Keywords: saturated formations
zbMATH Keywords: supplemented subgroups
zbMATH Keywords: complemented subgroups
zbMATH Keywords: Sylow subgroups
author: Wenbin Guo
MaRDI profile type: Publication
full work available at URL: s00229-008-0194-7
OpenAlex ID: W2013916113
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DOI: 10.1007/S00229-008-0194-7
description: scientific article; zbMATH DE number 5377117