- #1

Punkyc7

- 420

- 0

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

Any help would e greatly appreciated.

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- 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

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

Any help would e greatly appreciated.

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- #2

- 3,486

- 257

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.

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.

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.

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.

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.

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