On three-Engel groups

Abstract

The following conditions for a group G are investigated:(i) maximal class n subgroups are normal,(ii) normal closures of elements have nilpotency class n at most,(iii) normal closures are n–Engel groups,(iv) G is an (n+1 )-Engel group.Each of these conditions is a consequence of the preceding one. The second author has shown previously that all conditions are equivalent for n = 1. Here the question is settled for n = 2 as follows: conditions (ii), (iii) and (iv) are equivalent. The class of groups defined by (i) is not closed under homomorphisms, and hence (i) does not follow from the other conditions.