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Math Help - Number of Homomorphisms from Zn to Zm

  1. #1
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    Number of Homomorphisms from Zn to Zm

    Let $Hom$\{\mathbb{Z}_n, \mathbb{Z}_m\} denote the group of homomorphisms between \mathbb{Z}_n and \mathbb{Z}_m. Show that |$Hom$\{\mathbb{Z}_n, \mathbb{Z}_m\}|=$gcd$(m,n).

    The only thing I have is that since we're dealing with a homomorphism that |\phi(a)| divides |a|, but I don't see how that fact gets used to prove the above statement.
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  2. #2
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    Quote Originally Posted by mathematicalbagpiper View Post
    Let $Hom$\{\mathbb{Z}_n, \mathbb{Z}_m\} denote the group of homomorphisms between \mathbb{Z}_n and \mathbb{Z}_m. Show that |$Hom$\{\mathbb{Z}_n, \mathbb{Z}_m\}|=$gcd$(m,n).

    The only thing I have is that since we're dealing with a homomorphism that |\phi(a)| divides |a|, but I don't see how that fact gets used to prove the above statement.
    for an f \in $Hom$\{\mathbb{Z}_n, \mathbb{Z}_m\} suppose f(1+n\mathbb{Z})=k + m \mathbb{Z}. then f(r+n\mathbb{Z})=kr + m\mathbb{Z}. for all r. show that f is well-defined if and only if \frac{m}{d} \mid k, where d=\gcd(m,n).
    Last edited by NonCommAlg; September 9th 2010 at 05:40 PM.
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