We can just think of

.

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.