Finite groups with some \(\mathcal H\)-subgroups.
type: Item
instance of: scholarly article
title: Finite groups with some \(\mathcal H\)-subgroups.
zbMATH Open document ID: 1236.20015
author: Xianhua Li
author: Tao Zhao
author: Yong Xu
published in: Indagationes Mathematicae. New Series
publication date: 2011-09-22T00:00:00Z
review text: Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is called an \(\mathcal H\)-subgroup if \(N_G(H)\cap H^g\leq H\) for all \(g\) in \(G\). This subgroup embedding property was introduced by \textit{M. Bianchi, A. Gillio Berta Mauri, M. Herzog} and \textit{L. Verardi} [J. Group Theory 3, No. 2, 147-156 (2000; Zbl 0959.20024)] to characterize finite soluble groups in which normality is a transitive relation. The main results of the paper under review solve some open questions related to this subgroup embedding property, showing that if the family of the subgroups of the same order of the Sylow subgroups are \(\mathcal H\)-subgroups, then the structure of the group is quite restricted. Theorem~3.2 is typical: Let \(\mathfrak F\) be a saturated formation containing the class of all supersoluble groups, and let \(E\) be a normal subgroup of a group \(G\) containing the \(\mathfrak F\)-residual of \(G\). Then the group \(G\) belongs to \(\mathfrak F\) if every non-cyclic Sylow subgroup of \(E\) has a family of subgroups of the same order which are \(\mathcal H\)-subgroups of \(G\).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5949317
zbMATH Keywords: finite groups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: nilpotency
zbMATH Keywords: supersolubility
zbMATH Keywords: saturated formations
zbMATH Keywords: subgroup embedding properties
reviewed by: Adolfo Ballester-Bolinches
MaRDI profile type: Publication
full work available at URL: j.indag.2011.03.001
OpenAlex ID: W1502914407
cites work: Onp-Nilpotence and Supersolvability of Finite Groups
cites work: Finite Groups Whose Minimal Subgroups arec-Supplemented
cites work: On finite solvable groups in which normality is a transitive relation
cites work: On Supersolvable Groups and the Nilpotator
cites work: Finite soluble groups
cites work: Strongly closed 2-subgroups of finite groups
cites work: The influence of ℋ-subgroups on the structure of finite groups
cites work: On minimal subgroups of finite groups
cites work: On minimal non-PE-groups
cites work: Finite groups with NE-subgroups
cites work: On weakly \(s\)-permutable subgroups of finite groups.
DOI: 10.1016/J.INDAG.2011.03.001
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description: scientific article; zbMATH DE number 5949317