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)?