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On three-Engel groups

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.