Criteria for commutativity in large groups

In this paper we prove the following: 1. Let m greater than or equal to 2, n greater than or equal to 1 be integers and let G be a group such that (XY)(n) = (YX)(n) for all subsets X, Y of size m in G. Then a) G is abelian or a BFC-group of finite exponent bounded by a function of m and n. b) If m greater than or equal to n then G is abelian or \G\ is bounded by a function of m and n. 2. The only non-abelian group G such that (XY)(2) = (YX)(2) for all subsets X, Y of size 2 in G is the quaternion group of order 8. 3. Let m, n be positive integers and G a group such that [GRAPHICS] for all subsets X(i) of size m in C. Then G is n-permutable or \G\ is bounded by a function of m and n.