On (n+1/2)-Engel groups

Let n be a positive integer. We say that a group G is an (n+1/2)-Engel group if it satisfies the law [ x, y n, x ] = 1. The variety of (n+1/2)-Engel groups lies between the varieties of n-Engel groups and (n+1) -Engel groups. In this paper, we study these groups, and in particular, we prove that all (4+1/2)-Engel-groups are locally nilpotent. We also show that if G is a (4+1/2)-Engel p-group, where p ≥ 5 is a prime, then Gp is locally nilpotent.