Finite groups with weakly \(s\)-semipermutable subgroups.
type: Item
instance of: scholarly article
title: Finite groups with weakly \(s\)-semipermutable subgroups.
zbMATH Open document ID: 1256.20019
DOI: 10.4171/RSMUP/126-5
author: Changwen Li
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2012-02-10T00:00:00Z
full work available at URL: https://eudml.org/doc/241445
full work available at URL: http://rendiconti.math.unipd.it/volumes/vol126.php
review text: A subgroup \(H\) of a finite group \(G\) is said to be \(S\)-permutable in \(G\) if \(HS\) is a subgroup of \(G\) for all Sylow subgroups \(S\) of \(G\). This embedding property was introduced by Kegel, it has been extensively investigated and it turns out to be important in the structural study of the finite groups. An interesting variation of the \(S\)-permutability is the \(S\)-semipermutability which is defined as the permutability of a subgroup \(H\) with the Sylow subgroups corresponding to the primes not dividing its order. This subgroup embedding property has been considered in some recent papers dealing with problems concerning transitivity and subnormality. In this paper the author considers an interesting variation of \(S\)-semipermutability motivated by the property of c-supplementation introduced in the paper [\textit{A. Ballester-Bolinches, Y. Wang} and \textit{X. Guo}, Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)]: a subgroup \(H\) of a finite group \(G\) is said to be weakly \(s\)-semipermutable in \(G\) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\) is \(S\)-semipermutable in \(G\). This property is then used in order to derive information on the internal structure of a group \(G\) provided that various subgroups of \(G\) are weakly \(s\)-semipermutable in \(G\). The formation theory provides a wonderful framework to unify and generalise earlier results.
reviewed by: Adolfo Ballester-Bolinches
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6005071
zbMATH Keywords: finite groups
zbMATH Keywords: semipermutable subgroups
zbMATH Keywords: Sylow permutability
zbMATH Keywords: semipermutability
zbMATH Keywords: s-semipermutability
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: factorized groups
MaRDI profile type: Publication
cites work: Q4200361
cites work: Sylow-Gruppen und Subnormalteiler endlicher Gruppen
cites work: Influence of \(\pi\)-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group
cites work: On maximal subgroups of Sylow subgroups of inite groups
cites work: Q5298069
cites work: Finite groups with some subgroups of Sylow subgroups \(c\)-supplemented
cites work: A generalisation of Kramer's theorem and its applications
cites work: Cover-avoidance properties and the structure of finite groups
cites work: On weakly \(s\)-permutable subgroups of finite groups.
cites work: Q2762626
cites work: Q5577154
cites work: Finite soluble groups
cites work: \({\mathfrak h}\)-normalizers and local definitions of saturated formations of finite groups
cites work: Finite groups with some \(c\)-normal minimal subgroups
cites work: \(C\)-normality of groups and its properties
cites work: C-Supplemented subgroups of finite groups
cites work: A note on minimal subgroups of finite groups
cites work: Q4435419
cites work: On \(c\)-supplemented minimal subgroups of finite groups.
cites work: The influence of minimal subgroups on the structure of finite groups
cites work: On \(c\)-normality of certain subgroups of prime power order of finite groups.
cites work: The influence of c-normality of subgroups on the structure of finite groups II*
cites work: ON c-NORMAL MAXIMAL AND MINIMAL SUBGROUPS OF SYLOW SUBGROUPS OF FINITE GROUPS
cites work: Some notes on the minimal subgroups of Fitting subgroups of finite groups
cites work: Complementarity of Subgroups and the Structure of Finite Groups
cites work: Influence of normality on maximal subgroups of Sylow subgroups of a finite group
cites work: On complemented subgroups of finite groups
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description: scientific article; zbMATH DE number 6005071