\(\mathfrak{F}_{\tau}\)-embedded and \(\mathfrak{F}_{\tau\phi}\)-embedded subgroups of finite groups
type: Item
instance of: scholarly article
title: \(\mathfrak{F}_{\tau}\)-embedded and \(\mathfrak{F}_{\tau\phi}\)-embedded subgroups of finite groups
zbMATH Open document ID: 1352.20011
author: N. E. Zubov
published in: Algebra and Logic
publication date: 2016-01-05T00:00:00Z
review text: In the paper and in this review, all groups mentioned are finite and \(G\) stands for a finite group. Recall that a formation \(\mathfrak{F}\) is a class of groups such that it is closed under taking quotients and subdirect products. A formation is said to be saturated (respectively, solubly saturated) if \(G\in \mathfrak{F}\) whenever \(G/\Phi(G)\in\mathfrak{F}\) (respectively, \(G/\Phi(N)\in\mathfrak{F}\) for a soluble normal subgroup \(N\) of \(G\)). A chief factor \(L/K\) of \(G\) is \textit{\(\mathfrak{F}\)-central} if \((L/K)\rtimes (G/\mathrm{C}_G(L/K))\in\mathfrak{F}\). A normal subgroup \(N\) of \(G\) is said to be \(\mathfrak{F}\)-hypercentral (respectively, \(\mathfrak{F}\Phi\)-hypercentral) in \(G\) if either \(N=1\) or every chief factor (respectively, every non-Frattini chief factor) of \(G\) below \(N\) is \(\mathfrak{F}\)-central in \(G\). The \(\mathfrak{F}\)-hypercentre \(\mathrm{Z}_{\scriptstyle \mathfrak{F}}(G)\) (respectively, the \(\mathfrak{F}\Phi\)-hypercentre \(\mathrm{Z}_{\mathfrak{\scriptstyle F}\scriptstyle\Phi}(G)\)) of \(G\) is the product of all \(\mathfrak{F}\)-hypercentral (respectively, \(\mathfrak{F}\Phi\)-hypercentral) normal subgroups of~\(G\). A subgroup \(H\) of a group \(G\) is said to be quasinormal or permutable (respectively, S-quasinormal or S-permutable) in \(G\) if \(H\) permutes with all subgroups (respectively, all Sylow subgroups) of~\(G\). The influence on the structure of a group of the fact that some \(p\)-subgroups of a group satisfy a certain subgroup embedding property has been one of the major areas of interest in abstract group theory during the last years. Several properties have been introduced and studied. The aim of this paper is to give a common presentation of some known subgroup embedding properties and to unify some result that have parallel statements for the different embedding properties. Given a nonempty formation \(\mathfrak{F}\), a subgroup embedding functor \(\tau\) and a \(p\)-subgroup \(H\) of…
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6526401
zbMATH Keywords: finite group
zbMATH Keywords: subgroup functor
zbMATH Keywords: \(\mathfrak{F}_\tau\)-embedded subgroup
zbMATH Keywords: \(\mathfrak{F}_{\tau\Phi}\)-embedded subgroup
zbMATH Keywords: supersoluble group
zbMATH Keywords: formation
reviewed by: R. Esteban-Romero
MaRDI profile type: Publication
full work available at URL: s10469-015-9343-8
OpenAlex ID: W1872801267
cites work: Products of finite groups.
cites work: Sufficient conditions for supersolubility of finite groups
cites work: \(C\)-normality of groups and its properties
cites work: Finite groups with given \(s\)-embedded and \(n\)-embedded subgroups.
cites work: On solubility and supersolubility of some classes of finite groups.
cites work: Finite groups with \(sn\)-embedded or \(s\)-embedded subgroups
cites work: On ℱs-Quasinormal Subgroups of Finite Groups
cites work: On ℱ-Quasinormal Primary Subgroups of Finite Groups
cites work: Classes of Finite Groups
cites work: Finite soluble groups
cites work: A note on the Jordan-Hölder theorem
cites work: The embedding of quasinormal subgroups in finite groups
cites work: Sylow-Gruppen und Subnormalteiler endlicher Gruppen
cites work: Subgroups permutable with all Sylow subgroups
cites work: On factorizations of finite groups with ℱ-hypercentral intersections of the factors
cites work: On \(\mathcal F\)-supplemented subgroups of finite groups.
cites work: On the \(\mathcal X\Phi\)-hypercentre of finite groups.
cites work: On \(\mathcal F\Phi^*\)-hypercentral subgroups of finite groups.
cites work: Multiply \(\mathfrak L\)-composition formations of finite groups
cites work: Q5577154
cites work: Q3659962
cites work: On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups
cites work: Q4087401
cites work: M-groups and 2-groups
cites work: Automorphisms of quaternion-free 2-groups
cites work: Indices of elements and normal structure of finite groups.
cites work: On 𝓕-subnormal subgroups and Frattini-like subgroups of a finite group
cites work: Minimal non F-groups
cites work: On \(\Pi\)-property and \(\Pi\)-normality of subgroups of finite groups.
cites work: Some permutability properties related to \(\mathfrak F\)-hypercentrally embedded subgroups of finite groups
cites work: Subgroup lattices of groups
cites work: The influence of SS-quasinormality of some subgroups on the structure of finite groups.
cites work: On the partial \(\Pi\)-property of subgroups of finite groups.
cites work: Q3358930
cites work: Q5298069
cites work: On weakly \(\tau\)-quasinormal subgroups of finite groups.
cites work: On weakly \(S\)-embedded and weakly \(\tau\)-embedded subgroups.
cites work: On NearlyS-Permutable Subgroups of Finite Groups
cites work: Finite groups with systems of \(\Sigma\)-embedded subgroups.
cites work: A Generalization of Cover-Avoiding Properties in Finite Groups
cites work: Conditionally permutable subgroups and supersolubility of finite groups.
cites work: Q3544711
DOI: 10.1007/S10469-015-9343-8
Recommended article: On \(\sigma\)-embedded and \(\sigma\)-\(n\)-embedded subgroups of finite groups
Recommended article: On \(\pi\mathfrak F\)-embedded subgroups of finite groups.
Recommended article: On \(m\)-\(\sigma\)-embedded subgroups of finite groups
Recommended article: On \(n\)-\(\sigma\)-embedded subgroups of finite groups
Recommended article: Q5371470
Recommended article: Finite groups with given \(\sigma\)-embedded and \(\sigma\)-\(n\)-embedded subgroups
Recommended article: On S*-embedded subgroups of finite groups
Recommended article: Finite groups with \(H_{\mathcal{L}}\)-embedded subgroups
Recommended article: Finite groups with given nearly SΦ-embedded subgroups
Recommended article: Finite groups with systems of \(\Sigma\)-\( \mathfrak{F} \)-embedded subgroups
description: scientific article; zbMATH DE number 6526401