A Generalization of a Question Asked by B. H. Neumann

Let w∈ F2 be a word and let m and n be two positive integers. We say that a finite group G has the wm,n-property if however a set M of m elements and a set N of n elements of the group is chosen, there exist at least one element x∈ M and at least one element y∈ N such that w(x, y) = 1. Assume that there exists a constant γ< 1 such that whenever w is not the identity in a finite group X, then the probability that w(x1, x2) = 1 in X is at most γ. If m≤ n and G satisfies the wm,n-property, then either w is the identity in G or |G| is bounded in terms of γ, m and n. We apply this result to the 2-Engel word.