I'm reading the chapter in my abstract algebra book about homomorphisms between two groups. It states as a theorem:

Let f : G --> H be a homomorphism. Then,

(i) The kernel of f is a normal subgroup of G.

(ii) The range of f is a subgroup of H.

I don't understand part (ii). How could that not work? The function f is defined to be from G to H, and being a homomorphism the function must be surjective but not necessarily injective. So it seems to me that the range of f has to be exactly the group H.

My only possible guess is that homomorphisms don't have to be surjective? Say the function from Z6 to Z3:

0 1 2 3 4 5

0 1 2 0 1 2

Here, every element of Z3 has at least one correspondence in Z6. Are they saying this might not always be true for one group that is a homomorphic image of another?