Some open questions in the theory of generalized permutable subgroups.
type: Item
instance of: scholarly article
title: Some open questions in the theory of generalized permutable subgroups.
zbMATH Open document ID: 1193.20021
author: Wenbin Guo
author: Baojun Li
author: Fengyan Xie
published in: Science in China. Series A
publication date: 2010-01-06T00:00:00Z
review text: All groups considered in the review are finite. A subgroup \(H\) of a group \(G\) is said to be s-permutable in \(G\) if it permutes with every Sylow subgroup of \(G\). \textit{A. N. Skiba} introduced [in J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)] the following generalizations of this concept: 1) A subgroup \(H\) of a group \(G\) is weakly s-permutable in \(G\) if there exists a subnormal subgroup \(T\) such that \(G=HT\) and \(T\cap H\subseteq H_{sG}\), where \(H_{sG}\) is the largest s-permutable subgroup of \(G\) contained in \(H\). 2) A subgroup \(H\) of a group \(G\) is said to be weakly s-supplemented in \(G\) if \(H\) has a supplement \(T\) in \(G\) such that \(H\cap T\leqslant H_{sG}\). In the above cited paper the author proved two theorems concerning the belonging of a group \(G\) to a saturated formation \(\mathcal F\) containing all supersoluble groups, under the assumption that certain subgroups of any non-cyclic Sylow subgroups of some normal subgroups of \(G\) are weakly s-permutable in \(G\). Next he stated as open questions whether the condition of weak s-permutability in those results could be replaced by the condition of weak s-supplementation in both results. In the paper under review the authors construct an example showing that both open questions have negative answers. Moreover, they obtain some positive results under some additional hypotheses regarding the weak s-supplementation of certain maximal (respectively, minimal) subgroups of Sylow subgroups.
reviewed by: Ana Martínez-Pastor
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5653990
zbMATH Keywords: finite groups
zbMATH Keywords: generalized permutable subgroups
zbMATH Keywords: saturated formations
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: weakly s-supplemented subgroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: minimal subgroups
zbMATH Keywords: weakly s-permutable subgroups
zbMATH Keywords: subnormal subgroups
MaRDI profile type: Publication
full work available at URL: s11425-009-0045-3
OpenAlex ID: W2055305087
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DOI: 10.1007/S11425-009-0045-3
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description: scientific article; zbMATH DE number 5653990