Schur multipliers and the Lazard correspondence.
type: Item
instance of: scholarly article
title: Schur multipliers and the Lazard correspondence.
zbMATH Open document ID: 1315.20013
author: B. Eick
author: Max Horn
author: Seiran Zandi
published in: Archiv der Mathematik
publication date: 2012-11-21T00:00:00Z
review text: Let \(G\) be a finite \(p\)-group of nilpotency class less than \(p\), and let \(L\) be a Lie ring of \(p\)-power order of nilpotency class less than \(p\). The authors study properties behaving regularly under the Lazard correspondence \(G\mapsto L=\mathrm{Lie}(G)\mapsto\mathrm{Grp}(L)=G\). Throughout the paper it is assumed that the nilpotency class is less than \(p-1\). The main results are as follows. Schur multipliers of \(G\) and \(\mathrm{Lie}(G)\) are isomorphic as abelian groups. Isomorphism classes of Schur-covers of \(G\) and \(\mathrm{Grp}(L)\) correspond to each other under Lazard correspondence. A group (Lie ring) is capable if it is a central factor of some group (Lie ring). The epicenter of a group (Lie ring) is the intersection of all normal subgroups (ideals) with capable factor. The authors also show that epicenters of \(G\) and \(\mathrm{Lie}(G)\) are isomorphic as abelian groups, and capability is invariant under Lazard correspondence. These results are applied to groups of order \(p^5\) and class 3. The paper is very well written.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 20C25
Mathematics Subject Classification ID: 19C09
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 6109273
zbMATH Keywords: Schur multipliers
zbMATH Keywords: Schur covers
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Lazard correspondence
zbMATH Keywords: capable groups
zbMATH Keywords: capable Lie rings
zbMATH Keywords: epicenter
MaRDI profile type: Publication
full work available at URL: s00013-012-0426-7
OpenAlex ID: W2025130904
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cites work: Sur les groupes nilpotents et les anneaux de Lie
cites work: Q4858090
cites work: Multiplicators and groups with finite central factor-groups
DOI: 10.1007/S00013-012-0426-7
description: scientific article; zbMATH DE number 6109273