Propermutable characterizations of finite soluble \(PST\)-groups and \(PT\)-groups.
type: Item
instance of: scholarly article
title: Propermutable characterizations of finite soluble \(PST\)-groups and \(PT\)-groups.
zbMATH Open document ID: 1336.20023
DOI: 10.1134/S003744661502010X
author: Xiaolan Yi
published in: Siberian Mathematical Journal
publication date: 2016-01-06T00:00:00Z
review text: For \(H,X\) subgroups of \(G\) we say that \(H\) is \(X\)-propermutable in \(G\) if there is a subgroup \(B\) of \(G\) such that \(G=N_G(H)B\) and \(H^xB=BH^x\) for some \(x\in X\); for \(X=1\) the groups are just called propermutable. \(H\) is completely propermutable if \(H\) is propermutable in every \(K\) with \(H\subseteq K\subseteq G\). The author shows: (i) The soluble group \(G\) is a PST-group iff all Hall subgroups and all of their maximal subgroups are \(Z_\infty(G)\)-propermutable in \(G\). (ii) A soluble group of odd order is a PT-group iff all Hall subgroups and all subnormal subgroups of \(G\) are completely propermutable in \(G\) (Theorems A, B).
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 6526921
zbMATH Keywords: finite groups
zbMATH Keywords: completely propermutable subgroups
zbMATH Keywords: PST-groups
zbMATH Keywords: PT-groups
zbMATH Keywords: Hall subgroups
reviewed by: Hermann Heineken
MaRDI profile type: Publication
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full work available at URL: s003744661502010x
OpenAlex ID: W2464209961
description: scientific article; zbMATH DE number 6526921