Note that any permutation writable as a 3-cycle is an even permutation thus clearly the subgroup generated by all three cycles is a subgroup of . To show the other inclusion we first note that any may be written as the product of an even number of transpositions. It suffices then to show that the product of two distinct transpositions can be written as a 3-cycle. To do this we break it into two cases:
1. The transpositions aren't disjoin, in other words we have . But a little work shows that the above is actually equal to
2. The transpositions are disjoint, or we have . Again, with a little work, we can show that this is equivalent to
This finishes the proof.
What have you done for the second one?