Suppose that $f: G \rightarrow S_C$ be the action of Rubik's group $G$ on the collection of $C$ of corner cubelets. So if $C_1$ and $C_2$ be distinct corner cubelets. Show that there is a move $M \in G$ which maps FRU (front right up) to $C_1$ and FRD (front right down) to $C_2$. Then find a move $N$ in rubiks group $G$ which exchanges the cubelets FRU and FRD but leaves all the other corner cubelets in their place.
I am really lost on the first part of this problem. And I think for the second part, I must find a commutators in order to leave the other corner cubelets in one place with just exchanging FRU and FRD. Any help would be appreciated.
For the first part, first show it for the case when $C_1$ and $C_2$ are neighbors (this is physically obvious if you fiddle around with an actual cube, moving the FRU-FR-FRD stack around as a unit; how much you prose you need to describe that as a proof is a good question).
Then show that for each possible relative position between $C_1$ and $C_2$, there is a single rotation which makes $C_1$ and $C_2$ neighbors. They can then be moved to FRU/FRD as described above.
The wording of the exercise can sound like you're supposed to use the result of the first part to solve the second, but I think that would be a misunderstanding -- there's no obvious way the first part would be helpful for the second. Instead the point is probably that you're supposed later to combine these two parts to conclude that every transposition of corners can be made (as a conjugation of the FRU-FRD exchange).
Commutators alone will not do you any good in this part, because a commutator is always an even permutation and the one you're trying to create is odd.
Observe, however that if you start by making an F quarter-turn, what you need to reach a state with FRU and FRD exchanged is just a cyclic permutations of some of the front corners. Do you know how to construct 3-cycles as commutators between two permutations whose supports intersect at a single element only?