Let f:G--->G' be a homomorphism of group (G, *) onto (G', *) with kernal K. Let H' be a subgroup of G'. Let H={x in G such that f(x) is in H"}. Prove H is a subgroup of G and that K is a subset of H.

What I know:

definition of homomorphism: for all a, b in G, f(a*b)=f(a)*f(b)

properties of homomorphisms: f(e)=e', where e is in G and e' is in G' and f(x^{-1}) = (f(x))^{-1 }

Also, the kernal, K={a in G such that f(a) = e'}

To prove H is a subgroup of G, I need to show H is non-empty, closed, that * is associative in H, H has an identity, and H has inverses.

closed: for all a, b in H, a*b is in H

associative: since * is associative in G, it's associative in H

identity: there exists and e in H such that for all a in H a*e=e*a=a

inverses: for all a in H, aa^{-1}=a^{-1}a=e

I don't know how to start...please help.