Prove that two disjoint cycles commute
Let be two disjoint cycles. Clearly the which are fixed by both are fixed by the product, in which ever order it is taken. Now consider those which are not fixed by both. Since and are disjoint, for a fixed which is not fixed by both, exactly one of or moves . So, without loss of generality, let with . Since is part of the cycle , it must not be part of the cycle because and are disjoint. Then and , so both and are the same function on .