Tonio, is that the only way to go. Actually, i did this but i am not sure whether i am right or wrong.
I actually gave a one-one map from S->R. Does that solve this by any means?
Em, as for your question the yeah, their images must always be a 5-cycle in S_5
However, what you have proven will always hold although the result you are trying to prove does not hold. You have proven and you want to show , but as there exist counter-examples.
Let A be the image of f such that A=<(a_1, a_2, a_3, a_4, a_5)> and let B be the image of g such that B=<(b_1, b_2, b_3, b_4, b_5)>. Since the generator of A and B have the same cycle type, they are in the same orbit under the action of conjugation. That means, there exists such that , where b and a are generators of B and A, respectively.
Once we find a , the bijection is established between B and A. Think of a relabels the generator of B to the generator A. Then each element of B corresponds to A in a well-defined manner.