Some properties of ultra Hall and Schur pairs
type: Item
instance of: scholarly article
title: Some properties of ultra Hall and Schur pairs
zbMATH Open document ID: 1011.20032
author: Mohammad Reza R. Moghaddam
author: Ali Reza Salemkar
author: Mohammad Reza Rismanchian
published in: Archiv der Mathematik
publication date: 2002-05-21T00:00:00Z
review text: Let \(\Phi\) be a set of words of a free group, then \(\Phi(G)\) and \(\Phi^*(G)\) denote the verbal and marginal subgroups of \(G\) corresponding to \(\Phi\). If \(\mathcal K\) is a class of groups, then \((\Phi,{\mathcal K})\) is said to be a `Schur pair' (`weak Schur pair') if, for every group \(G\), \(G/\Phi^*(G)\in\mathcal K\) (\(G/\Phi^*(G)\in\mathcal K\) and \(G/\Phi(G)\in\mathcal K\)) implies \(\Phi(G)\in\mathcal K\). Let \(\Phi M(G)\) denote the Baer invariant of \(G\), then the first result of this paper shows that \((\Phi,{\mathcal K})\), where \({\mathcal K}=PQS_n\mathcal K\), is a Schur pair if and only if \(G\in\mathcal K\) implies \(\Phi M(G)\in\mathcal K\). The concept of an ultra Hall pair is introduced and it is shown that, if \({\mathcal K}=NPQS_n\mathcal K\) and \((\gamma_2,{\mathcal K})\) is a Schur pair, then \((\phi,{\mathcal K})\) is an ultra Hall pair, for every outer commutator word \(\phi\). Corollaries of both theorems are given.
Mathematics Subject Classification ID: 20E10
zbMATH DE Number: 1742824
zbMATH Keywords: marginal subgroups
zbMATH Keywords: Schur pairs
zbMATH Keywords: verbal subgroups
zbMATH Keywords: Baer invariants
zbMATH Keywords: ultra Hall pairs
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
full work available at URL: s00013-002-8222-4
OpenAlex ID: W1983808910
DOI: 10.1007/S00013-002-8222-4
Recommended article: SOME PROPERTIES ON THE SCHUR MULTIPLIER OF A PAIR OF LIE ALGEBRAS
Recommended article: Some remarks on generalized Schur pairs
Recommended article: Some properties on the Schur multiplier of a pair of groups.
Recommended article: Q5435108
Recommended article: Q5405546
Recommended article: Q4533258
Recommended article: Q3475948
Recommended article: Variants of theorems of Schur, Baer and Hall
Recommended article: Q3708540
Recommended article: Schur Multipliers of Cartan Pairs
description: scientific article; zbMATH DE number 1742824