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**1. Homework Statement**

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

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

**2. Homework Equations**

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

**3. The Attempt at a Solution**

I know that since [itex]\nu_1[/itex] and [itex]\nu_2[/itex] are epimorphisms, they are surjective homomorphisms. So [itex]Im(\nu_1)=G_1/H_1[/itex] and [itex]Im(\nu_2)=G_2/H_2[/itex]. 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 [itex]\sigma^{\prime} : H_1 \rightarrow H_2[/itex] such that [itex]\iota_2 \circ \sigma^{\prime} = \sigma \circ \iota_1[/itex] if and only if [itex]\sigma[H_1] \subset H_2[/itex]. If such a [itex]\sigma^{\prime}[/itex] exists, it is a homomorphism.

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