Finite groups in which normality or quasinormality is transitive.
type: Item
instance of: scholarly article
title: Finite groups in which normality or quasinormality is transitive.
zbMATH Open document ID: 1079.20023
published in: Archiv der Mathematik
publication date: 2005-01-20T00:00:00Z
review text: A `T-group' is a group in which normality is a transitive relation and a `PT-group' is one in which permutability is transitive. Recently there has been considerable interest in the problem of characterizing finite T-groups, especially in the soluble case. The characterizations usually take the form of restrictions on the Sylow structure of the group. In the work under review the author gives several new criteria for a finite group to be a soluble T-group or PT-group. The novelty here is that the restrictions only apply to Sylow \(p\)-subgroups of a group \(G\) for which the prime \(p\) divides the order of the generalized Fitting subgroup \(F^*(G)\). We quote two of the results obtained. (i)~A finite group \(G\) is a soluble T-group if and only if each \(p\)-subgroup is pronormal in \(G\) for all \(p\) dividing \(|F^*(G)|\). (ii)~A finite group \(G\) is a soluble PT-group if and only if \(H\leq P\in\text{Syl}_p(G)\) implies that \(H\) is permutable in \(N_G(P)\) for all \(p\) dividing \(|F^*(G)|\). (A subgroup \(H\) is `pronormal' in \(G\) if \(H\) and \(H^g\) are conjugate in \(\langle H,H^g\rangle\) for all \(g\in G\).)
reviewed by: Derek J. S. Robinson
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 2130536
zbMATH Keywords: finite solvable groups
zbMATH Keywords: solvable T-groups
zbMATH Keywords: solvable PT-groups
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: transitive normality
zbMATH Keywords: transitive quasinormality
zbMATH Keywords: generalized Fitting subgroup
zbMATH Keywords: Sylow subgroups
author: M. Asaad
MaRDI profile type: Publication
full work available at URL: s00013-004-1065-4
OpenAlex ID: W2032177310
DOI: 10.1007/S00013-004-1065-4
description: scientific article; zbMATH DE number 2130536