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Math Help - Finding all the homomorphisms

  1. #1
    Newbie
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    Oct 2009
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    Finding all the homomorphisms

    Find all the homomorphisms from Z -> Z x Z and Z x Z -> Z.

    For the first, I've found and proven the following.

    \phiZ -> Z x Z such that a -> (a,0) or (0,a) or (0,0).

    For the second one:

    \phiZ x Z -> Z such that (a,b)-> a or b (the "forgetful" function) or 0.

    So I have found three. Am I on the right track? I have verified that these are indeed a homomorphism.
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  2. #2
    Junior Member
    Joined
    Feb 2007
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    There are still others, for example, \phi(a) = (2a, 3a).

    Instead of just trying to come up with examples of homomorphisms, begin with the assumption that  \phi: \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z} is a homomorphism and see what you can deduce from that assumption alone.
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