I'm working on the following qual problem and I'm not sure how to do parts b and c.
a) This uses a nifty argument with considering the map $N_G(S) \to \operatorname{Aut}(S)$ via conjugation. And using the fact that $|N_G(S)/C_G(S)|$ is odd, and divides the order of $\operatorname{Aut}(S)$ so it must be 1.
b) Not sure how to proceed after the hint. I think we're supposed to use strong Cayley's theorem here by saying that the kernel of this homomorphism is the largest normal subgroup of $G$ contained in $S$.
c) No idea how to do this one, but I was thinking if we can show the subgroup has order n(from part b) then we note that $|N||S| = |G|$ and so $NS = G$.