P-groups and homomorphisms...

P-groups and homomorphisms...

If $f: G \rightarrow H$ is a surjective homomorphism, $G$ finite, prove
the following:
1) If $P$ is a p-subgroup of $G$, then $f(P)$ is a p-subgroup of $H$.
2) If $S$ is a Sylow-p-subgroup of $G$, then then $f(S)$ is a
Sylow-p-subgroup of $H$.
EDIT: I have been doing this problem for several days now on and off and I
have no idea how to start...I'd rather skip the hints honestly, as I've
been given a few and still am missing a key piece.