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Algebra - morphisms

  1. Jan 3, 2012 #1
    1. The problem statement, all variables and given/known data

    List all morphisms of addition for Z6 -> Z3. (integers mod 6 to integers mod 3)

    2. Relevant equations

    Definition of morphism in text:

    A morphism f:(X,*) -> (X',*') is defined to be a function on X to X' which carries the operation * on X into the operation *' on X', in the sense that

    f(x*y)=(fx)*'(fy)

    for all x,y in X.

    A morphism of addition is where * and *' are operations of addition.

    3. The attempt at a solution

    I've been working ahead in my class and I'm just not sure I found all the morphisms and for some reason I had trouble with this. I can do the proofs later on in the exercises though..., so I think I'm okay with the definition of a morphism...

    The only function that comes to mind that satisfies what was required above is f:Z6->Z3 defined by:

    f(x) = remainder after division by 3.

    I can't seem to really be certain that this is the only one though or if there are others. If it is the only one, then what if I Z3=X and Z6=X'. That is, f:Z3 -> Z6. If it isn't the only one, what approach could I employ to find it?

    What I did was simply write down the definition and do a few examples to see what jumped out at me to define f as.

    Thanks!

    Edit:
    The book hasn't covered groups yet, so I don't know. And I don't know why I didn't go through the individual options, haha. Okay, I'll try looking at it that way.
     
    Last edited: Jan 3, 2012
  2. jcsd
  3. Jan 3, 2012 #2

    micromass

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    OK, first of all I'll assume you mean morphisms of groups here.

    Here is a method how to find all the morphisms in this case:
    1) What must f(0) be??
    2) What possibilities can f(1) have??
    3) What does the choice of f(1) mean for f(2), f(3), etc. ??
    4) Does the choice of f(1) leave us with a well-defined morphism?? I.e. does f(6)=f(0)=?? Does f(3+3)=f(3)+f(3)?? Etc.
     
  4. Jan 3, 2012 #3
    Ahhhhh. Okay. Haven't finished yet, but I think I see how to reason it through now by my scratch work now. Thank you!
     
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