Without lose of generality say that . Let be a non-identity element with cycle decomposition where these cycles are non-identity cycles. The way you prove this is by exploring various cases. The first case is that are not found as entries in any one of the cycles and so has cycle decomposition and now apply the definition of the signum function. The second case is that are found as entries in the cycle but are found both in a single cycle (by relabeling if necessary). Therefore, we can think of as . Now and so and apply the definition of signum function. Another case is when are found as entries in the cycle by are found in different cycles say (by relabeling if necessary). So say that and now compute and apply the definition again. And finally there are cases when (but not both) appear as entries in one of the cycles. Just do all these cases and the proof ought to be complete. For the second part I imagine you need to use the result that every permutation is a product of transpositions.