Criteria for Commutativity in Large Groups

In this paper we prove the following:1.Let m ≥ 2, n ≥ 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. Thena)G is abelian or a BFC-group of finite exponent bounded by a function of m and n.b)If m ≥ 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 for all subsets Xi of size m in G. Then G is n-permutable or |G| is bounded by a function of m and n.