Having trouble on a couple of problems...know the definitions, not very good at applying them. i can show that $\displaystyle j_1 $ and $\displaystyle j_2$ 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 $\displaystyle i$ the identity in G and $\displaystyle i$' the identity in H. Define the functions as:

$\displaystyle j_1$: G $\displaystyle \rightarrow$ G x H such that $\displaystyle \pi_1$: g $\displaystyle \mapsto$(g, $\displaystyle i$')

$\displaystyle j_2$: H $\displaystyle \rightarrow$ G x H such that $\displaystyle \pi_2$: h $\displaystyle \mapsto$($\displaystyle i$, h)

Show that the functions $\displaystyle j_1 $ and $\displaystyle j_2$ are homomorphism.

What are the kernels of $\displaystyle j_1 $ and $\displaystyle j_2$?

What are the images of $\displaystyle j_1 $ and $\displaystyle j_2$?

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:

$\displaystyle \pi_1$: G x H $\displaystyle \rightarrow$ G such that $\displaystyle \pi_1$: (g, h) $\displaystyle \mapsto$ g

$\displaystyle \pi_2$: G x H $\displaystyle \rightarrow$ H such that $\displaystyle \pi_2$: (g, h) $\displaystyle \mapsto$ k

Show that the projection maps are surjective homomorphisms

What are the kernels of the $\displaystyle \pi_1$ and $\displaystyle \pi_2$?

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