ENGEL ELEMENTS IN GROUPS ALIREZA ABDOLLAHIDepartment of Mathematics, University of Isfahan, Isfahan 81746-73441, IRAN. Email:
[email protected] Abstract We give a survey of results on the structure of right and left Engel elements of a group. We also present some new results in this topic. 1 Introduction Let G be any group and x,y ∈ G. Define inductively the n-Engel left normed commutator [x,n y] = [x,y,...,y] n of the pair (x,y) for a given non-negative integer| {z }n, as follows: −1 −1 −1 y [x,0 y] := x, [x,1 y] := [x,y]= x y xy =: x x , and for all n> 0 [x,n y] = [[x,n−1 y],y]. An element a ∈ G is called left Engel whenever for every element g ∈ G there exists a non-negative integer n = n(g, a) possibly depending on the elements g and a such that [g,n a] = 1. For a positive integer k, an element a ∈ G is called a left k-Engel element of G whenever [g,k a] = 1 for all g ∈ G. An element a ∈ G is called arXiv:1002.0309v1 [math.GR] 1 Feb 2010 a bounded left Engel element if it is left k-Engel for some k. We denote by L(G), Lk(G) and L(G), the set of left Engel elements, left k-Engel elements, bounded left Engel elements of G, respectively. In definitions of various types of left Engel elements a of a group G, we observe that the variable element g appears on the left hand side of the element a.