ON CERTAIN WEAK ENGEL-TYPE CONDITIONS IN GROUPS

Let w(x, y) be a word in two variables and B the variety determined by w. In this paper we raise the following question: if for every pair of elements a, b in a group G there exists g is an element of G such that w(a(g) , b) = 1, under what conditions does the group G belong to B? In particular, we consider the n-Engel word w(x, y) = [x,(n) y]. We show that in this case the property is satisfied when the group G is metabelian. If n = 2, then we extend this result to the class of all solvable groups.