Onto Homomorphism: G/H Isomorphic to K

  • Thread starter Punkyc7
  • Start date
In summary, the conversation discusses the existence of a homomorphism between two groups G and K, with the kernel being H. The First Isomorphism Theorem is mentioned as a potential approach, but the speaker is struggling to find a solution. The question is posed about finding a surjective homomorphism with a kernel of H, and the possibility of composing it with the given isomorphism.
  • #1
Punkyc7
420
0
If G/H is [itex]\cong[/itex] K show there exist an homomorphism which is ont λ:G [itex]\rightarrow[/itex] K with the kernel of λ=HI am having a hard time figuring out what this should be. I have a feeling it is easy if you know how to look at it.

I have been trying the First Isomorphism Theorem but I can't seem to get.

Any help would e greatly appreciated.
 
Physics news on Phys.org
  • #2
You have an isomorphism [itex]\phi : G/H \rightarrow K[/itex]. Can you find a surjective homomorphism [itex]\theta : G \rightarrow G/H[/itex] which has ker [itex]\theta = H[/itex]? What happens when you compose [itex]\theta[/itex] with [itex]\phi[/itex]?
 

FAQ: Onto Homomorphism: G/H Isomorphic to K

1. What is an onto homomorphism?

An onto homomorphism is a function between two algebraic structures that preserves the operations of the structures and maps the entire domain to the entire range. In other words, every element in the target structure has at least one corresponding element in the source structure.

2. What does G/H Isomorphic to K mean?

G/H isomorphic to K means that there exists a bijective onto homomorphism from the quotient group G/H to the group K. In simpler terms, G/H and K are structurally equivalent and can be mapped onto each other in a way that preserves the group operations.

3. How can we prove that G/H is isomorphic to K?

To prove that G/H is isomorphic to K, we need to show that there exists a bijective onto homomorphism from G/H to K. This can be done by constructing a mapping that preserves the group operations and shows that every element in K has a corresponding element in G/H and vice versa.

4. What are the applications of onto homomorphism?

Onto homomorphism has various applications in mathematics, computer science, and physics. In mathematics, it is used to study and compare different algebraic structures. In computer science, it is used in encryption and coding theory. In physics, it is used to study symmetries in the physical laws.

5. Is every onto homomorphism an isomorphism?

No, not every onto homomorphism is an isomorphism. It is possible for a function to be onto and preserve the operations of the structures, but not be bijective, which is a necessary condition for an isomorphism. However, if an onto homomorphism is also injective, then it is an isomorphism.

Similar threads

Do Isomorphic Groups Have Isomorphic Centers?
Replies
3
Views
1K
Isomorphism between HK and H x K
Replies
19
Views
4K
An exercise with the third isomorphism theorem in group theory
Replies
5
Views
2K
Showing isomorphism between fractions and a quotient ring
Replies
1
Views
1K
Normal group of order 60 isomorphic to A_5
Replies
6
Views
1K
Isomorphism is an equivalence relation on groups
Replies
9
Views
8K
How Is the Quotient Group G/H Isomorphic to G'?
2
Replies
37
Views
4K
F is an isomorphism from G onto itself,...., show f(x) = x^-1
Replies
3
Views
1K
Compact Hausdorff Spaces: Star-Isomorphic Unital C*-Algebras
Replies
5
Views
1K
1st Isomorphism thm for dihedral gps
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top