How to show this is a homomorphism?

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In summary, the given homomorphism θ maps elements of the group Z6 to their corresponding remainders when divided by 2. This can be expressed as θ(x) = x mod 2. To show that this is a homomorphism, we can use the general form of elements in Z6 and Z2, and consider the remainders when divided by 2. The proof involves using the definition of homomorphism of groups, and may require writing out a few more steps if considering Z(mod6) and Z(mod2).
  • #1
SMA_01
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How to show this is a homomorphism?

Homework Statement



Let θ: Z6→Z2 be given by θ(x)=the remainder of x when divided by 2 (as in the division algorithm)

Homework Equations





The Attempt at a Solution


I am stuck, this is all I have:

Let m,n be in Z6
θ(m +6 n)...

I'm not sure how to proceed. Any help is appreciated. Thanks
 
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  • #2


If m is an element of Z6. What does it look like?
 
  • #3


@kru- What do you mean? I imagine Z6 to look like a circle that starts and ends at 6...not sure if this is very accurate though.
 
  • #4


I think kru_ is asking you how would you express an element of 6Z or 2Z or (any integer)Z in the general form (hint, odd integers are expressed as 2k + 1, where k is in Z).

From there, you should think what is the remainder of any element in 6Z divided by 2 and simply use the definition of homomorphism of groups.

If it is Z6 and Z2 (or Z(mod6) and Z(mod2) in other words) you are talking about, then you follow a very similar logic, just with a little more writing required in your proof.
 

FAQ: How to show this is a homomorphism?

1. What is a homomorphism?

A homomorphism is a function between two algebraic structures, such as groups or rings, that preserves the operations of the structures. In other words, the function maps elements from one structure to another in a way that respects the operations of the structures.

2. How can I show that a function is a homomorphism?

To show that a function is a homomorphism, you must demonstrate that it preserves the operations of the structures. This can be done by showing that for any inputs x and y in the domain of the function, the function applied to the operation of x and y is equal to the operation of the function applied to x and y. In other words, f(x * y) = f(x) * f(y), where * represents the operation in the structure.

3. What are the properties of a homomorphism?

There are several properties that a homomorphism must have. These include preserving the identity element, preserving inverses, and preserving the order of elements. Additionally, a homomorphism must be a well-defined function, meaning that each element in the domain has a unique image in the codomain.

4. Can a function be a homomorphism in one structure but not another?

Yes, a function may be a homomorphism in one structure but not another. This is because the operations and elements of different structures may behave differently, and a function that preserves the operations in one structure may not preserve them in another.

5. How is a homomorphism different from an isomorphism?

A homomorphism is a function that preserves the operations of a structure, while an isomorphism is a homomorphism that is also a one-to-one and onto function. This means that an isomorphism not only preserves the operations, but also preserves the structure and relationships between elements in the structures. In other words, an isomorphism is a bijective homomorphism.

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