Hall subgroups and \(\pi\)-separable groups
type: Item
instance of: scholarly article
title: Hall subgroups and \(\pi\)-separable groups
zbMATH Open document ID: 0889.20011
author: Zhaowei Du
published in: Journal of Algebra
publication date: 1998-04-02T00:00:00Z
review text: Let \(E_\pi\) denote the collection of groups that have Hall \(\pi\)-subgroups for a set \(\pi\) of primes. The author's abstract: Let \(G\) be a finite group and \(\pi\) a set of primes. In this paper, some new criteria for \(\pi\)-separable groups and \(\pi\)-solvable groups in terms of Hall subgroups are proved. The main results are the following: Theorem. \(G\) is a \(\pi\)-separable group if and only if (i) \(G\) satisfies \(E_\pi\) and \(E_{\pi'}\); (ii) \(G\) satisfies \(E_{\pi\cup(q)}\) and \(E_{\pi'\cup(p)}\) for all \(p\in\pi\), \(q\in\pi'\). Theorem. \(G\) is a \(\pi\)-separable group if and only if (i) \(G\) satisfies \(E_\pi\) and \(E_{\pi'}\); (ii) \(G\) satisfies \(E_{p,q}\) for all \(p\in\pi\) and \(q\in\pi'\).
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 1088611
zbMATH Keywords: Hall subgroups
zbMATH Keywords: \(\pi\)-separable groups
zbMATH Keywords: \(\pi\)-solvable groups
reviewed by: Homer F. Bechtell
MaRDI profile type: Publication
full work available at URL: jabr.1997.7034
OpenAlex ID: W2085171956
cites work: Q3919880
cites work: On finite factorizable groups
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DOI: 10.1006/JABR.1997.7034
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description: scientific article; zbMATH DE number 1088611