Finite groups with a seminormal Hall subgroup.
type: Item
instance of: scholarly article
title: Finite groups with a seminormal Hall subgroup.
zbMATH Open document ID: 1128.20011
published in: Mathematical Notes
publication date: 2007-05-21T00:00:00Z
review text: The paper studies finite groups with a seminormal Hall subgroup. A subgroup \(H\) of a group \(G\) is said to be seminormal in \(G\) if there is a subgroup \(K\) in \(G\) such that \(HK=G\) and the product \(HK_1\) is a proper subgroup of \(G\) for any subgroup \(K_1\) of \(K\) which differs from \(K\). In this case, we refer to the subgroup \(K\) as a supercomplement to \(H\) in \(G\). The main results proved in the paper are the following: (1) If \(G\) is a finite group having a seminormal Hall \(\pi\)-subgroup \(H\), then the group \(G\) is \(\pi\)-solvable in each of the following cases: (i) \(H\) is 2-nilpotent (i.e., \(H\) has a normal Hall \(2'\)-subgroup); (ii) \(H\) is solvable and \(3\not\in\pi\). (2) If every noncyclic Sylow subgroup of a group \(G\) is seminormal, then the group \(G\) is solvable and its third commutator subgroup is nilpotent.
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5155560
zbMATH Keywords: finite groups
zbMATH Keywords: Hall subgroups
zbMATH Keywords: seminormal subgroups
zbMATH Keywords: solvability
zbMATH Keywords: nilpotency
zbMATH Keywords: solvable groups
zbMATH Keywords: supersolvable groups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: Sylow subgroups
author: V. S. Monakhov
reviewed by: Rodica Covaci
MaRDI profile type: Publication
full work available at URL: s11006-006-0173-9
OpenAlex ID: W1996243352
cites work: Produkte nilpotenter Gruppen
cites work: Subgroups of prime power index in a simple group
cites work: Q3744347
cites work: Q3934684
DOI: 10.1007/S11006-006-0173-9
description: scientific article; zbMATH DE number 5155560