Guided proof to the isomorphism theorems.

  • Thread starter *melinda*
  • Start date
  • #1
86
0

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.


Homework Equations



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


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:

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
9,395
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 on Guided proof to the isomorphism theorems.

  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
3
Views
281
  • Last Post
Replies
19
Views
2K
Top