Spreads and nilpotence class in nilpotent groups and Lie algebras.
type: Item
instance of: scholarly article
title: Spreads and nilpotence class in nilpotent groups and Lie algebras.
zbMATH Open document ID: 1320.20022
author: Avinoam Mann
published in: Journal of Algebra
publication date: 2014-11-07T00:00:00Z
review text: For a non-abelian finite \(p\)-group \(G\), let \(p^{b(G)}\) be the maximum, and \(p^{s(G)}\) the minimum, of sizes of conjugacy classes of non-central elements of \(G\). The number \(\delta=\delta(G)=b(G)-s(G)\) is called the spread of \(G\). \textit{A. Jaikin-Zapirain} proved [Proc. Am. Math. Soc. 133, No. 10, 2817-2820 (2005; Zbl 1077.20032)] that the nilpotency class of \(G\) is at most \(2\delta^2+2\delta+3\). In the present paper this bound in improved to \(\delta p/(p-1)+3-1/(p-1)\leq 2\delta+2\). Similar parameters can be introduced for finite-dimensional nilpotent Lie algebras over a field, with codimensions of centralizers in place of exponents in class sizes, as well as for finitely generated torsion-free nilpotent groups, and previous bounds for the nilpotency class are also significantly improved in these settings.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 6366940
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: nilpotent Lie algebras
zbMATH Keywords: torsion-free nilpotent groups
zbMATH Keywords: nilpotency classes
zbMATH Keywords: conjugacy class sizes
zbMATH Keywords: orders of centralizers
zbMATH Keywords: dimensions of centralizers
zbMATH Keywords: spreads
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.08.013
OpenAlex ID: W2011683862
Wikidata QID: Q115350904
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DOI: 10.1016/J.JALGEBRA.2014.08.013
description: scientific article; zbMATH DE number 6366940