Left 3-Engel elements in groups of exponent 5.
type: Item
instance of: scholarly article
title: Left 3-Engel elements in groups of exponent 5.
zbMATH Open document ID: 1306.20041
author: Gunnar Traustason
published in: Journal of Algebra
publication date: 2014-09-02T00:00:00Z
review text: An element \(a\) of a group \(G\) is called left \(3\)-Engel if \([x,a,a,a]=1\) for all \(x\in G\) or equivalently \(\langle a,a^x\rangle\) is nilpotent of class at most \(2\) for all \(x\in G\). The main question about left \(3\)-Engel elements is the following Question. Does every left \(3\)-Engel element of a group lie in the Hirsch-Plotkin radical of the group? The author of the paper under review starts a systematic approach to the latter question. It is observed that a left \(3\)-Engel element \(a\) of a group \(G\) lies in the Hirsch-Plotkin radical of \(G\) if and only if \(K(a,n)=\langle a,a^{g_1},a^{g_2},\ldots,a^{g_n}\rangle\) is nilpotent for all \(n\) and all elements \(g_1,\ldots,g_n\). Let \(a\) be a left \(3\)-Engel element. Then the group \(K(a,n)\) has the property that \(\langle x,y^g\rangle\) is nilpotent of class at most \(2\) for all \(x,y\in\{a,a^{g_1},\ldots,a^{g_n}\}\) and all \(g\in G\). This is the motivation for the definition of sandwich groups: a group \(K\) is called a sandwich group if \(K=\langle X\rangle\) and \(\langle x,y^g\rangle\) is nilpotent of class at most \(2\) for all \(x,y\in X\) and \(g\in K\). The author proves that \(3\)-generated sandwich groups are nilpotent of class at most \(5\). The author first considers certain groups generated by three elements \(a,b,c\) such that \(\langle a,b\rangle\), \(\langle a,c\rangle\) and \(\langle b,c\rangle\) are nilpotent of classes \(r\), \(s\), \(t\), respectively. The latter groups are called of type \((r,s,t)\). The structure of the free group of type \((1,2,2)\) is completely determined by giving a power-commutator presentation. Next the structure of the free group of latter type with the extra condition that \(c\) is a left \(3\)-Engel element is determined. Then, the structure of the free group of type \((1,2,3)\) with extra conditions that \([b,c,c]=1\) and \(c\) is left \(3\)-Engel is determined. The proofs consist of tedious commutator calculus, however the author has trie…
reviewed by: Alireza Abdollahi
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 6338798
zbMATH Keywords: Engel elements
zbMATH Keywords: groups of exponent \(5\)
zbMATH Keywords: left \(3\)-Engel elements
zbMATH Keywords: Hirsch-Plotkin radical
zbMATH Keywords: sandwich groups
zbMATH Keywords: power-commutator presentations
zbMATH Keywords: nilpotent groups
describes a project that uses: NQ
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.05.017
OpenAlex ID: W2047792279
cites work: Left 3-Engel elements in groups.
cites work: On the Right and Left 4-Engel Elements
cites work: Engelsche Elemente Noetherscher Gruppen
cites work: Groups that are Pairwise Nilpotent
cites work: The Engel elements of a soluble group
cites work: Refined solvable presentations for polycyclic groups
cites work: THE FREE BURNSIDE GROUPS OF SUFFICIENTLY LARGE EXPONENTS
cites work: Infinite Burnside groups of even exponent
cites work: A NOTE ON THE LOCAL NILPOTENCE OF 4-ENGEL GROUPS
cites work: Engel-4 Groups of Exponent 5
cites work: SOLUTION OF THE RESTRICTED BURNSIDE PROBLEM FOR GROUPS OF ODD EXPONENT
cites work: Q3362124
DOI: 10.1016/J.JALGEBRA.2014.05.017
description: scientific article; zbMATH DE number 6338798