Eventually all the elements will be associated with one cycle or another but each with one cycle only. It can be shown that representation of a permutation as a product of cycles is unique up to their order.
Any permutation is a product of transpositions.
As before, this means that f produces the same effect as consecutive application of a series of transpositions. The product is not symmetric and the transpositions are performed from right to left. We can apply this argument to all the cycles g and then execute all thus obtained transpositions sequentially. Representation of a permutation as a product of cycles is unique. Its representation as a product of transpositions is not.