# Thread: homomorphism, projection map, kernels, and images

1. ## homomorphism, projection map, kernels, and images

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)?

2. You are just be confused by the notation, dont care about the notation!
try to understand the map, what it really is. what is the variable? and what is fixed element?
You can take G=H=R(real number group with respect to addition) as a example to help you get the idea of the definition and maps.

3. Shanks: Thank you for your input.

I finally cleared up what was with the two different arrows...one with the bar and one without. Now, I am, again, stuck on the proof. Here's what I got so far:

Ok.

I showed that the functions and are homomorphism by using the fact that the identity in G and ' the identity in H.

The kernel of is { } and the kernel of is {' }.

The image of is ( h,' ) and the image of is ( ,k).

Based on the Fund. Homomo. Thm, we can tell that G and H are subgroups of G x H.

For the projection maps:

The kernel of is image of which is ( h,' ), and the kernel of is the image of which is ( ,k).

We can now say that H and K are normal subgroups.

I'm getting hung up on proving the surjective homomorphism part. How do you prove something is onto? I know for a homomorphism you show that:
$\displaystyle \phi\,\!$(ab) = $\displaystyle \phi\,\!$(a)$\displaystyle \phi\,\!$(b)

Any more help would be much appreciated.

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# projection homomorphism definition

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