Let m n be positive integers and be a class of groups. We say that a group G satisfies the condition m n, if for every two subsets M and N of cardinalities m and n, respectively, there exist x ∈ M and y ∈ N such that x y ∈ . In this article, we study groups G satisfies the condition m n, where is the class of nilpotent groups. We conjecture that every infinite m n-group is weakly nilpotent (i.e., every two generated subgroup of G is nilpotent). We prove that if G is a finite non-soluble group satisfies the condition m n, then G ≤ maxm nc2maxmn2 logmaxmn 60 !, for some constant c (in fact c ≤ maxm n). We give a sufficient condition for solubility, by proving that a m n-group is a soluble group whenever m +n < 59. We also prove the bound 59 cannot be improved and indeed the equality for a nonsoluble group G holds if and only if G A5, the alternating group of degree 5.
