On conjugate-permutable subgroups of finite groups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1288.20014
DOI: 10.1142/S0219498813500606
author: A. A. Heliel
author: T. M. Al-Gafri
publication date: 2013-09-11T00:00:00Z
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 6207518
zbMATH Keywords: finite groups
zbMATH Keywords: complete sets of Sylow subgroups
zbMATH Keywords: conjugate-permutable subgroups
zbMATH Keywords: \(\mathfrak Z\)-permutable subgroups
zbMATH Keywords: \(C\)-\(\mathfrak Z\)-permutable subgroups
zbMATH Keywords: \(p\)-nilpotent groups
zbMATH Keywords: supersoluble groups
zbMATH Keywords: generalised Fitting subgroup
zbMATH Keywords: saturated formations
zbMATH Keywords: maximal subgroups
zbMATH Keywords: subgroup embedding properties
MaRDI profile type: Publication
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published in: Journal of Algebra and its Applications
title: On conjugate-permutable subgroups of finite groups.
review text: The groups in the paper under review and in this report are assumed to be finite. The study of the influence of the maximal subgroups of the Sylow subgroups of a distinguished normal subgroup on the structure of a group has motivated a lot of interest during the last years in the research in abstract group theory. This paper is a new contribution in this research line. Here a complete set \(\mathfrak Z\) of Sylow subgroups of a finite group \(G\) is considered, that is, for each prime \(p\) dividing the order of \(G\), \(\mathfrak Z\) contains exactly one Sylow \(p\)-subgroup \(G_p\) of \(G\). Given a non-empty subset \(C\) of \(G\), a subgroup \(H\) of \(G\) is said to be \(C\)-\(\mathfrak Z\)-permutable (conjugate-\(\mathfrak Z\)-permutable) subgroup of \(G\) if there exists some \(x\in C\) such that \(H^xG_p=G_pH^x\) for all \(G_p\in\mathfrak Z\). In all this report, \(\mathfrak Z\) will denote a complete set of Sylow subgroups of a group \(G\) and \(C\) will be a soluble normal subgroup of \(G\). In addition, \(\mathfrak F\) will denote a saturated formation containing the class \(\mathfrak U\) of all supersoluble groups. Recall that a saturated formation is a class of groups closed under quotients, subdirect products, and Frattini extensions.NEWLINENEWLINE Theorem~3.1 states that if \(p\) is the smallest prime number dividing the order of \(G\) and the maximal subgroups of \(G_p\in\mathfrak Z\) are \(C\)-\(\mathfrak Z\)-permutable subgroups of \(G\), then \(G\) is \(p\)-nilpotent. In Theorem~3.2, it is proved that \(G\in\mathfrak F\) if and only if there exists a normal subgroup \(H\) in \(G\) such that \(G/H\in\mathfrak F\) and the maximal subgroups of \(G_p\cap H\) are \(C\)-\(\mathfrak Z\)-permutable subgroups of \(G\) for all \(G_p\in\mathfrak Z\). In Theorem~3.5, it is shown that \(G\in\mathfrak F\) if and only if there exists a soluble normal subgroup \(H\) of \(G\) such that \(G/H\in\mathfrak F\) and the maximal subgroups of the Sylow subgroups of the Fi…
reviewed by: R. Esteban-Romero
description: scientific article; zbMATH DE number 6207518