Some sufficient conditions for a finite group to be supersolvable.
type: Item
instance of: scholarly article
title: Some sufficient conditions for a finite group to be supersolvable.
zbMATH Open document ID: 1260.20032
published in: Acta Mathematica Hungarica
publication date: 2013-01-02T00:00:00Z
review text: The authors extend the results of \textit{Ya. G. Berkovich} [Mat. Sb., N. Ser. 74(116), 75-92 (1967); translation in Math USSR, Sb. 3(1967), 69-83 (1969; Zbl 0183.02901)] and \textit{M. Asaad} [Commun. Algebra 38, No. 10, 3616-3620 (2010; Zbl 1205.20026)] by proving the following. Theorem~1.1. Let \(G\) be a group of odd order. Let \(G=G_1G_2\cdots G_n\) be the product of pairwise permutable subgroups \(G_1,G_2,\dots,G_n\). If the Sylow \(p\)-subgroups of all \(G_i\) are cyclic, then \(G\) is \(p\)-supersolvable. Theorem~1.3. Let \(G\) be a group of even order. Let \(G=G_1G_2\cdots G_n\) be the product of pairwise permutable subgroups \(G_1,G_2,\dots,G_n\) such that \(G_2G_3\cdots G_n\) is of odd order and \(G_1G_i\) is a mutually \(m\)-permutable product (\(i=2,3,\dots,n\)), that is, \(G_1\) permutes with every maximal subgroup of \(G_i\) and \(G_i\) permutes with every maximal subgroup of \(G_1\). If the Sylow subgroups of all \(G_i\) are cyclic, then \(G\) is supersolvable.
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 6121083
zbMATH Keywords: supersolvable grous
zbMATH Keywords: \(p\)-supersolvable groups
zbMATH Keywords: permutable subgroups
zbMATH Keywords: groups of odd order
zbMATH Keywords: products of subgroups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: mutually permutable products
zbMATH Keywords: supersolvability
zbMATH Keywords: maximal subgroups
author: M. Asaad
author: V. S. Monakhov
MaRDI profile type: Publication
full work available at URL: s10474-011-0170-7
OpenAlex ID: W2066676956
cites work: A Condition for the Supersolvability of Finite Groups
cites work: ON PRODUCTS OF FINITE SUPERSOLUBLE GROUPS
cites work: ON SOLVABLE GROUPS OF FINITE ORDER
cites work: On supersolvability of finite groups
cites work: Solvability of groups of odd order
cites work: Endliche Gruppen I
cites work: Q3125641
cites work: Q3949183
DOI: 10.1007/S10474-011-0170-7
description: scientific article; zbMATH DE number 6121083