Determine is endomorphism, kernel, epimorphism, monomorphism

• Shackleford
In summary, the conversation revolved around the correctness of the solutions for problems 1b and 1h, which involve transformation phi from the multiplicative group G to itself. The identity for both cases is 1, and it was deemed unnecessary work to check for monomorphism directly as the kernel being {1} is sufficient proof. Additionally, the definition of an endomorphism was discussed in relation to the problem, and it was suggested to prove it using the definition of a group homomorphism.
Shackleford
I'm reasonably certain I did 1b correctly. I'm not sure about 1h. In both cases, since phi is the transformation from the multiplicative group G to the multiplicative group G, the identity is 1.

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110729_200311.jpg?t=1311988303

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110729_200408.jpg?t=1311988318

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110729_200704.jpg?t=1311988336

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Seems to be entirely correct!

One small detail, though. You checked that 1b was a monomorphism directly with the definition. This is unnecessary work as you also calculated the kernel. Knowing that the kernel is {1} is good enough for showing it's a mono.

Yup, looks good.

micromass said:
Seems to be entirely correct!

One small detail, though. You checked that 1b was a monomorphism directly with the definition. This is unnecessary work as you also calculated the kernel. Knowing that the kernel is {1} is good enough for showing it's a mono.

Thanks for checking my work.

How does the kernel = {1} show one-to-one?

SammyS said:
Yup, looks good.

Thanks.

According to the definition, a homomorphism is an endomorphism if the mapping is to itself, i.e. G = G'. So, in this problem, since G = G (R - {0}), you just have to show it's a homomorphism or not.

Shackleford said:
How does the kernel = {1} show one-to-one?

You could try showing this! It only requires the definition of a group homomorphism. And it's actually both a necessary and sufficient condition (it'll be an "if and only if" proof).

1. What is an endomorphism?

Endomorphism refers to a map or function that maps a mathematical object to itself. In other words, it is a function that takes an object and returns the same type of object.

2. What is the kernel of an endomorphism?

The kernel of an endomorphism is the set of all elements in the domain that are mapped to the identity element in the codomain. It is essentially the set of all elements that are mapped to zero or the neutral element.

3. What is an epimorphism?

An epimorphism is a map or function that is surjective, meaning that every element in the codomain is mapped to by at least one element in the domain. In other words, every element in the output has at least one input that produces it.

4. What is the difference between a monomorphism and an epimorphism?

A monomorphism is a map or function that is injective, meaning that each element in the codomain has at most one corresponding element in the domain. In other words, each output has at most one input that produces it. The main difference between a monomorphism and an epimorphism is the directionality of the mapping. A monomorphism focuses on the uniqueness of the input, while an epimorphism focuses on the surjectivity of the output.

5. How are endomorphisms, kernels, epimorphisms, and monomorphisms related?

Endomorphisms can have kernels, which are the set of elements that are mapped to the identity element. Epimorphisms and monomorphisms are types of endomorphisms, with the former being surjective and the latter being injective. These concepts are all related to the idea of mappings between mathematical objects and understanding how elements in one set are related to elements in another set.

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