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, b ∈ G 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 a ∈ G is said to be a left Engel element if for all g ∈ G there exists a positive integer n = n(g) such that en(g, a) = 1. If instead one can for all g ∈ G 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.