Let m, n be positive integers. We denote by R(m,n) (respectively P(m,n)) the class of all groups G such that, for every n subsets $X_1,X_2ldots,X_n$, of size m of G there exits a non-identity permutation $sigma$ such that $X_1X_2{cdots}X_n{cap}X_{sigma(1)}X_{/sigma(2)}{cdots}X_{/sigma(n)} eqphi$ (respectively $X_1X_2{cdots}X_n=X_{/sigma(1)}X_{sigma(2)}{cdots}X_{sigma(n)}$). Let G be a non-abelian group. In this paper we prove that (i) $G{in}P$(2,3) if and only if G isomorphic to $S_3$, where $S_n$ is the symmetric group on n letters. (ii) $G{in}R$(2, 2) if and only if ${mid}G{mid}geq8$. (iii) If G is finite, then $G{in}R$(3, 2) if and only if ${mid}G{mid}geq14$ or G is isomorphic to one of the following: SmallGroup(16, i), $iin$ {3, 4, 6, 11, 12, 13}, SmallGroup(32, 49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of order m in the GAP [13] library.