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Homework Help: How to show this is a homomorphism?

  1. Mar 22, 2012 #1
    How to show this is a homomorphism???

    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    3. 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
  2. jcsd
  3. Mar 22, 2012 #2
    Re: How to show this is a homomorphism???

    If m is an element of Z6. What does it look like?
  4. Mar 22, 2012 #3
    Re: How to show this is a homomorphism???

    @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.
  5. Mar 22, 2012 #4


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    Re: How to show this is a homomorphism???

    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.
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