Sorry I am not used to using the tex code, but will learn and explain in words for now! I am trying to prove that for all distinct x and y in a finite set X, there exists a function f in P(X) (the permutation group) such that f(x)=x' and f(y)=y'. Note: x' and y' are also distinct. I have started by letting P(X) = the symmetry group and have shown that there exists a cycle (x y) which takes two distinct values and returns 2 distinct values. I am pretty sure x can equal y'. Now I am stuck! I have been told that a case-by-case analysis will probably be required. Can somebody point me in the right direction?