• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Guided proof to the isomorphism theorems.

  • Thread starter *melinda*
  • Start date
86
0
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.
 
Last edited:

matt grime

Science Advisor
Homework Helper
9,394
3
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.
 

Related Threads for: Guided proof to the isomorphism theorems.

Replies
1
Views
2K
  • Posted
Replies
4
Views
2K
  • Posted
Replies
7
Views
3K
  • Posted
Replies
3
Views
2K
  • Posted
Replies
5
Views
2K
  • Posted
Replies
5
Views
2K
  • Posted
Replies
2
Views
2K
Replies
19
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top