On \(\mathcal M\)-permutable Sylow subgroups of finite groups.
type: Item
instance of: scholarly article
title: On \(\mathcal M\)-permutable Sylow subgroups of finite groups.
zbMATH Open document ID: 1214.20022
author: Long Miao
author: Wolfgang Lempken
published in: Illinois Journal of Mathematics
publication date: 2010-12-08T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1290435341
review text: Let \(G\) be a finite group and \(p\) a prime divisor of \(|G|\). A nontrivial \(p\)-subgroup \(P\) of \(G\) is called \(\mathcal M\)-permutable in \(G\) if there exists a set \(\mathcal M_d(P)=\{P_1,\dots,P_d\}\) of maximal subgroups \(P_i\) of \(P\) and a subgroup \(B\) of \(G\) such that \[ \bigcap^d_{i=1}P_i=\Phi(P)\quad\text{and}\quad|P:\Phi(P)|=p^d\tag{1} \] and \[ G=PB\quad\text{and}\quad P_iB=BP_i<G\quad\text{for any }P_i\text{ of }\mathcal M_d(P).\tag{2} \] If \(p\) is an odd prime divisor of the order of \(G\) and \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(G\) is \(p\)-nilpotent if and only if \(N_G(P)\) is \(p\)-nilpotent and \(P\) is \(\mathcal M\)-permutable in \(G\) (Theorem 3.1). If \(\mathcal F\) is a saturated formation containing all supersoluble groups and \(H\) is a normal subgroup of \(G\) such that \(G/H\in\mathcal F\) and every noncyclic Sylow subgroup of \(H\), \(F(H)\) or \(F^*(H)\) is \(\mathcal M\)-permutable in \(G\), then \(G\in\mathcal F\) (Theorem 3.2, 3.4 and 3.6., respectively).
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5824936
zbMATH Keywords: finite groups
zbMATH Keywords: permutable subgroups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: saturated formations
zbMATH Keywords: supersoluble groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5824936