Homomorphisms & the isomorphism thm.

P: G to H is a group homomorphism, L = Image(P) = {h in H | h= P(g) for some g in G}.

A. Show L is a subgroup of H.

Now assume G= under addition and H= .

B. Show that P can't be onto. Which subgroups of are possible for L?

C. Find a specific example of some group homomorphism P from G to H which isn't the trivial map.

my thoughts:

for part A, I had that P(g)P(h) = P(gh) for any g, h in G, P( )= and P( )= P , by the definition of a hom, but was told that this is wrong because I need to discuss L. What am I missing here?

for B, I know the possible subgroups of H are <1>, <(123)>, <(12)>,<(23)>, <(13)>, and H, all of order 1, 2, 3 or 6, which all divide the order of G -- from there I don't know how to show there is no onto hom.

for (C), I have no idea how to find a mapping that will work!

Thank you so much for any help!