ENSURING A GROUP IS WEAKLY NILPOTENT

Let m, n be positive integers and X be a class of groups. We say that a group G satisfies the condition X(m, n), if for every two subsets M and N of cardinalities m and n, respectively, there exist x is an element of M and y is an element of N such that < x, y > is an element of X. In this article, we study groups G satisfies the condition R(m, n), where R is the class of nilpotent groups. We conjecture that every infinite R(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 R(m, n), then vertical bar G vertical bar <= max{m, n}c(2max{m, n}2) [log(60)(max{m,n})]!, for some constant c ( in fact c <= max{m, n}). We give a sufficient condition for solubility, by proving that a R(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 non-soluble group G holds if and only if G congruent to A(5), the alternating group of degree 5.