Guided proof to the isomorphism theorems.

[*melinda*]

Homework Statement

Let G_1 and G_2 be groups with normal subgroups H_1 and H_2, respectively. Further, we let \iota_1 : H_1 \rightarrow G_1 and \iota_2 : H_2 \rightarrow G_2 be the injection homomorphisms, and \nu_1 : G_1 \rightarrow G_1/H_1 and \nu_2 : G_2/H_2 be the quotient epimorphisms.

Given that there exists a homomorphism \sigma : G_1 \rightarrow G_2, show that there exists a unique mapping \overline{\sigma} : G_1/H_1 \rightarrow G_2/H_2 such that \overline{\sigma} \circ \nu_1 = \nu_2 \circ \sigma if and only if \sigma[H_1] \subset H_2. If such a \overbar{\sigma} exists, it is a homomorphism.

Homework Equations

There aren't any equations, as this is a proof.

The Attempt at a Solution

I know that since \nu_1 and \nu_2 are epimorphisms, they are surjective homomorphisms. So Im(\nu_1)=G_1/H_1 and Im(\nu_2)=G_2/H_2. But I really don't see how to get this proof off the ground. Please help get me started.

The next question reads as follows.

Prove that there exists a unique mapping \sigma^{\prime} : H_1 \rightarrow H_2 such that \iota_2 \circ \sigma^{\prime} = \sigma \circ \iota_1 if and only if \sigma[H_1] \subset H_2. If such a \sigma^{\prime} exists, it is a homomorphism.

 

[matt grime]

Let s=sigma, v=v_1 and w=v_2, cos I want to do this without having to type in tex. The only place to start is with the composite ws, since that is a well defined map, and it goes from G_1 to G_2/H_2. This gives a map of G_1/H_1 if and only if H_1 is in the kernel of the map ws. Which is if and only if... That is existence. Uniqueness we'll come to in a second.