Subnormal, permutable, and embedded subgroups in finite groups.
type: Item
instance of: scholarly article
title: Subnormal, permutable, and embedded subgroups in finite groups.
zbMATH Open document ID: 1245.20016
published in: Central European Journal of Mathematics
publication date: 2012-01-16T00:00:00Z
review text: A subgroup \(H\) of a group \(G\) is (a) permutable (S-permutable), if it is permutable with all subgroups (all Sylow subgroups) of \(G\), (b) semipermutable (S-semipermutable) if it is permutable with all subgroups (all Sylow subgroups) of coprime order of \(G\), (c) seminormal (S-seminormal), if it is normalized by all subgroups (all Sylow subgroups) of coprime order of \(G\). The purpose of this article is to study these subgroup embedding properties and to compare them with other permutability properties like those considered by \textit{A. Ballester-Bolinches} and \textit{R. Esteban-Romero} [J. Algebra 251, No. 2, 727-738 (2002; Zbl 1010.20013)]. The following result (Theorem 1.5) is an example: For a solvable group \(G\) the following statements on all of its subnormal subgroups are equivalent: (i) they are seminomal, (ii) they are semipermutable, (iii) they are S-semipermutable, (iv) they are permutable.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5997974
zbMATH Keywords: finite solvable groups
zbMATH Keywords: permutable subgroups
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: semipermutable subgroups
zbMATH Keywords: seminormal subgroups
zbMATH Keywords: PST-groups
zbMATH Keywords: Sylow subgroups
author: J. C. Beidleman
author: M. F. Ragland
reviewed by: Hermann Heineken
MaRDI profile type: Publication
full work available at URL: s11533-011-0029-8
OpenAlex ID: W2084402135
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DOI: 10.2478/S11533-011-0029-8
description: scientific article; zbMATH DE number 5997974