On Certain Weak Engel-Type Conditions in Groups

Abstract

Let w x y be a word in two variables and 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 ∈ G such that w a g b = 1, under what conditions does the group G belong to ? 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.