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Math Help - Homomorphisms

  1. #1
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    Homomorphisms

    Find ALL group homomorphisms from f: Z/2Z --> Z/4Z.

    I know elements of Z/2Z have order 1 or 2.
    I know elements of Z/4Z have order 1, 2, or 4.

    I believe I must use this to find all the homomorphism. I don't know what to do from here. Any advice? Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Homomorphisms

    Quote Originally Posted by jzellt View Post
    Find ALL group homomorphisms from f: Z/2Z --> Z/4Z.

    I know elements of Z/2Z have order 1 or 2.
    I know elements of Z/4Z have order 1, 2, or 4.

    I believe I must use this to find all the homomorphism. I don't know what to do from here. Any advice? Thanks
    The maps are determined by where 1 gets mapped to. Try to prove that 1\mapsto x implies |x|\mid 2. Moreover, prove that if |x|\mid 2 then 1\mapsto x gives a well-defined morphism.


    In geneneral, it's helpful to remember that as abelian groups \text{Hom}(\mathbb{Z}_m,\mathbb{Z}_n)\cong \mathbb{Z}_{(m,n)}.
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  3. #3
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    Re: Homomorphisms

    to amplify what mr. drexel is saying:

    we know f(0) = 0. so the only choice we have is where f sends 1.

    4 choices:

    f(1) = 0
    f(1) = 1
    f(1) = 2
    f(1) = 3.

    what goes wrong when we send 1-->1 (think: what does f(1+1) have to be)?
    how about 1-->3?
    does 1-->2 share these problems? 1-->0?

    *******

    or, one could ignore the "element" view, and just think about what we know about homomorphisms. f(Z/2Z) has to be a subgroup of Z/4Z. how many subgroups of Z/4Z exist? is it even possible for f to be surjective?

    also, ker(f) has to be a subgroup of Z/2Z. again, what are our possible choices? can you establish some kind of relationship between possible subgroups ker(f) of Z/2Z, and subgroups f(Z/2Z) of Z/4Z?
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