This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: "Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?" We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this

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... n can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.