Having trouble on a couple of problems...know the definitions, not very good at applying them. i can show that and are homomorphisms. im just lost on the relationship between homomorphism, kernels, and images and I think the notation is really throwing me off. also, since I'm not real sure of the notation, I can't tell if there's a typo in the first definition.

any help or guidance would be much appreciated.

thanks for looking!

Let G and H be groups with the identity in G and ' the identity in H. Define the functions as:

: G G x H such that : g (g, ')

: H G x H such that : h ( , h)

Show that the functions and are homomorphism.

What are the kernels of and ?

What are the images of and ?

Based on the Fundamental Homomorphism Thm, what can you tell of subgroups H x K

Let G and H be groups. Define the projection map:

: G x H G such that : (g, h) g

: G x H H such that : (g, h) k

Show that the projection maps are surjective homomorphisms

What are the kernels of the and ?

What more can be said about the subgroups mentioned above (H x K)?