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Engel groups

Published online by Cambridge University Press:  05 July 2011

Gunnar Traustason
Affiliation:
University of Bath
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

We give a survey on Engel groups with particular emphasis on the development during the last two decades.

Introduction

We define the n-Engel word en(x, y) as follows: e0(x, y) = x and en+1(x, y) = [en(x, y), y]. We say that a group G is an Engel group if for each pair of elements a, bG we have en(a, b) = 1 for some positive integer n = n(a, b). If n can be chosen independently of a, b then G is an n-Engel group.

One can also talk about Engel elements. An element aG is said to be a left Engel element if for all gG there exists a positive integer n = n(g) such that en(g, a) = 1. If instead one can for all gG choose n = n(g) such that en(a, g) = 1 then a is said to be a right Engel element. If in either case we can choose n independently of g then we talk about left n-Engel or right n-Engel element respectively.

So to say that a is left 1-Engel or right 1-Engel element is the same as saying that a is in the center and a group G is 1-Engel if and only if G is abelian. Every group that is locally nilpotent is an Engel group. Furthermore for any group G we have that all the elements of the locally nilpotent radical are left Engel elements and all the elements in the hyper-center are right Engel elements.

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Publisher: Cambridge University Press
Print publication year: 2011

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