New characterizations of finite supersoluble groups.
type: Item
instance of: scholarly article
title: New characterizations of finite supersoluble groups.
zbMATH Open document ID: 1161.20013
author: Baojun Li
author: A. N. Skiba
published in: Science in China. Series A
publication date: 2008-09-09T00:00:00Z
review text: Let \(A\) be a subgroup of \(G\) and let \(X\) be a non-empty subset of \(G\). \(A\) is called an \(X\)-semipermutable subgroup of \(G\) if \(A\) has a supplement \(T\) in \(G\) such that for every subgroup \(T_1\) of \(T\) there exists an element \(x\in X\) such that \(AT_1^x=T_1^xA\). One of the most remarkable results in this paper is the following theorem: If every Sylow subgroup of a group \(G\) is \(F(G)\)-semipermutable in \(G\), then \(G\) is supersoluble and \(G/O_{p,p'}(G)\) is a cyclic group of order \(p-1\) for any prime \(p\). Also, the authors obtain that a group \(G\) is supersoluble if and only if every maximal subgroup of every Sylow subgroup of \(G\) not having a supersoluble supplement in \(G\) is \(F(G)\)-semipermutable in \(G\).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 5323360
zbMATH Keywords: semipermutable subgroups
zbMATH Keywords: supplements
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: supersolubility
zbMATH Keywords: maximal subgroups
reviewed by: Francisco Pérez-Monasor
MaRDI profile type: Publication
full work available at URL: s11425-007-0155-8
OpenAlex ID: W2225766620
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DOI: 10.1007/S11425-007-0155-8
description: scientific article; zbMATH DE number 5323360