Finite groups with Hall \(\pi\)-subgroups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.20014
published in: Sbornik: Mathematics
publication date: 2012-07-31T00:00:00Z
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6062367
zbMATH Keywords: existence of Hall subgroups
zbMATH Keywords: conjugacy of Hall subgroups
zbMATH Keywords: finite groups
zbMATH Keywords: Hall property
zbMATH Keywords: Sylow theorems
zbMATH Keywords: local formations
MaRDI profile type: Publication
full work available at URL: sm2012v203n03abeh004225
OpenAlex ID: W2062178094
DOI: 10.1070/SM2012V203N03ABEH004225
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title: Finite groups with Hall \(\pi\)-subgroups.
author: V. A. Vedernikov
review text: Generalizing the known properties of Sylow subgroups, Hall introduced the classes \(E_\pi\), \(C_\pi\) and \(D_\pi\) of finite groups, [for a survey see \textit{E. P. Vdovin} and \textit{D. O. Revin}, Russ. Math. Surv. 66, No. 5, 829-870 (2011); translation from Usp. Mat. Nauk. 66, No. 3, 3-46 (2011; Zbl 1243.20027)]. The author substantially generalizes these classes.NEWLINENEWLINE Let \(\mathcal F\) be a nonempty class of finite groups. Let \(E_\pi(\mathcal F)\) be the class of finite groups which contain an \(S_\pi(\mathcal F)\)-subgroup, \(C_\pi(\mathcal F)\) be the class of finite groups which contain precisely one conjugacy class of \(E_\pi(\mathcal F)\)-subgroups, and \(D_\pi(\mathcal F)\) be the class of finite groups in which every \(\mathcal F_\pi\)-subgroup is contained in an \(S_\pi(\mathcal F)\)-subgroup.NEWLINENEWLINE Theorems generalizing classical results of P. Hall, Wielandt, Baer, and Hartley are obtained. Few excerpts. Let \(\mathcal F\) be a (Q,S,Ext)-closed class of finite groups. The finite group \(G\in E_\pi(\mathcal F)\) (\(G\in D_\pi(\mathcal F)\)) if and only if \(\Aut_G(H/K)\in E_\pi(\mathcal F)\) (\(\Aut_G(H/K)\in D_\pi(\mathcal F)\)) for every composition factor \(H/K\) of \(G\). \(E_\pi(\mathcal F)\) and \(C_\pi(\mathcal F)\) are local formations. An \(E_\pi(\mathcal F)\)-by-\(D_\pi(\mathcal F)\) group is a \(D_\pi(\mathcal F)\)-group.
reviewed by: János Kurdics
description: scientific article; zbMATH DE number 6062367