Groups nearly of prime exponent and nearly Engel Lie algebras
type: Item
instance of: scholarly article
title: Groups nearly of prime exponent and nearly Engel Lie algebras
zbMATH Open document ID: 0912.17003
author: Avinoam Mann
author: Consuelo Martínez
published in: Archiv der Mathematik
publication date: 1999-05-26T00:00:00Z
review text: Let \(L\) be a finite Lie algebra having prime characteristic \(p\) (which can be considered as a finite-dimensional Lie algebra over the prime field \(GF(p)\)). \(L\) is said to satisfy the Engel condition if \([\dots[[x,y],y],\dots,y]= x\text{ ad}(y)^n= 0\) for every \(x,y\in L\). The authors prove that if \(S\) is a vector subspace of \(L\) and if the probability \(\alpha\) that two elements in \(S\) satisfy the Engel identity \(E_n\) is greater than \((2^{n+1}- 1)/2^{n+1}\) then the Engel identity \(E_n\) is satisfied in \(S\). They also prove that if \(G\) is a finite \(p\)-group and that the probability that an element of \(G\) has order dividing \(p\) is \(>(3^p- 2)/(3^p- 1)\), then \(L(G)\), the Lie algebra associated to \(G\), satisfies the Engel condition \(E_{p-1}\).
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 1203764
zbMATH Keywords: Engel Lie algebras
zbMATH Keywords: groups nearly of prime exponent
zbMATH Keywords: Engel condition
zbMATH Keywords: Engel identity
reviewed by: Melih Boral
MaRDI profile type: Publication
Wikidata QID: Q115390140
full work available at URL: s000130050227
OpenAlex ID: W2041336891
DOI: 10.1007/S000130050227
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description: scientific article; zbMATH DE number 1203764