Abstract

In 1970 I.D. Macdonald exhibited a nilpotent group in which the square and the inverse of a right 3-Engel element need not be 3-Engel and thereby showing that the set of right 3-Engel elements of a group need not form a subgroup. In this note a nilpotent group for each n ≥ 3 is constructed such that the set of right n-Engel elements in each group is not a subgroup.

Introduction

An element a of a group is called a right n-Engel element, n a positive integer, if for each element g of the group [a, ng] = 1 (cf. [Rob72, p. 40]). Commutators are written left-normed and repeated entries in a commutator are indicated by left subscripts. Clearly, the set of right 1-Engel elements is the centre of the group and therefore a subgroup. W. Kappe [Kap61] proved that the set of right 2-Engel elements of a group is also a subgroup. I.D. Macdonald [Mac70] showed that the set of right 3-Engel elements of a group need not form a subgroup by constructing a group with an element that is right 3-Engel but whose inverse and square are not. In this paper we will construct a group for each n ≤ 3 with a right n-Engel element whose inverse and square are not right n-Engel. This answers a question raised at the conference by W. Kappe for such an example for n = 4.