Abstract Algebra Problem 2?

Question

Kind of a long one:

Lef f: G to H be a homomorphism, and suppose that the order of the ker(f) = n. For any h in f(G), find the number of elements in f ^-1(h). I got that part, heres the part I'm having trouble with.
Let f, G, and H be as in part a. Use your result in a to prove taht if K is a finite subgroup of G, then the order of f(K) divides the order of K.

Thanks

Answer 1

The previous result is that the number of elements in the preimage of each element in f(K) (or more generally in f(G)) is n. All those sets are disjoint and their union is K, so the total number of elements in K is n * (Number of elements in f(K)).
--> |K| is a multiple of |f(K)|

Another way of looking at is that if N is the kernel of f, f(K) is isomorphic to the quotient group K/N which always divides the order of K.