Almost Engel finite and profinite groups

Let g be an element of a group G. For a positive integer n, let En(g) be the subgroup generated by all commutators [:::[[x; g]; g]; : : : ; g] over x 2 G, where g is repeated n times. We prove that if G is a prfinite group such that for every g 2 G there is n = n(g) such that En(g) is finite, then G has afinite normal subgroup N such that G=N is locally nilpotent. The proof uses the Wilson{Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group G, we prove that if, for some n, jEn(g)j 6 m for all g 2 G, then the order of the nilpotent residual 1(G) is bounded in terms of m