Engel elements in groups (Q1336811)

The authors show that no power of a right \(n\)-Engel element of a group need be left \(n\)-Engel even if the group is nilpotent. Their main result is: for every \(n \geq 5\) there is a group which (i) has a right \(n\) Engel-element \(a\) and an element \(b\) such that the commutator \([b, na]\) has infinite order, (ii) is nilpotent of class \(n + 2\), (iii) is abelian- by-(nilpotent of class 3) and, (iv) is (nilpotent of class 2)-by-cyclic. The result was suggested by computer implementation of a nilpotent quotient algorithm on groups satisfying Engel conditions.