Finite Groups with a Subnormality Condition

Let m and n be positive integers. We study S(m, n), the class of all groups G that for all subsets M and N of G of sizes m and n, there exist x is an element of M and y E N such that (x) is subnormal in (x, y). In fact, if G is a finite group, then we find some sharp bounds for m and n such that G E S(m, n) implies that G is a nilpotent group. Also we find a bound depending only on m and n for the order of all nonnilpotent finite groups in S(m, n).