Consider the maps $\mu:\mathbb{Z}→\mathbb{Z}$ and $\mu:\mathbb{Z}→\mathbb{Z}_2$. For example if I am asked to find the number of homomorphisms of $\mathbb{Z}$ into $\mathbb{Z}$, and of $\mathbb{Z}$ onto $\mathbb{Z}_2$, what do I have to do? I don't have idea here. Thanks.
How to find the number of homomorphisms of $\mathbb{Z}$ into $\mathbb{Z}$?
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0@JuniorII To see Pedro's Remark: Note that any element in a group can be written in terms of its generators. Now, since the group $(\mathbb{Z}, +)$ is a cyclic ( and hence abelian) group and it has only on generator, you must be able to write any element as finite "sum" of its generator $1$. This together with $\mu$ being a homomorphism should prove his remark. – 2011-12-28
2 Answers
A group homomorphism $f:\mathbb{Z}\to R$ from $\mathbb{Z}$ to any commutative ring $R$ is determined by the image of $1$ since $f(n)=n f(1)$. If $f$ is required to be a ring homomorphism, we know $f(1)=1$, so there is only one homomorphism at all. This property means that $\mathbb{Z}$ is the "initial object" in the category of rings.
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0You guys were right of course. Sorry for the confusion there. Fixed it. – 2011-12-28
This question isn't very well-posed, but luckily there's a general answer to your question. The following identities are easily verified
1) $\text{Hom}(\mathbb{Z},\mathbb{Z})\cong\mathbb{Z}$
2) When $n\geqslant2$ $\text{Hom}(\mathbb{Z},\mathbb{Z}_n)\cong\mathbb{Z}_n$
3) When $n\geqslant 2$ $\text{Hom}(\mathbb{Z}_n,\mathbb{Z})\cong0$
4) When $n,m\geqslant2$ $\text{Hom}(\mathbb{Z}_n,\mathbb{Z}_m)\cong\mathbb{Z}_{(n,m)}$
With this all in mind, and using the fact that $\text{Hom}$ and $\oplus$ (for finitely many groups) commute in both variables one can deduce that
$\text{Hom}\left(\mathbb{Z}^r\times\mathbb{Z}_{\ell_1}\times\cdots\times\mathbb{Z}_{\ell_n},\mathbb{Z}^s\times\mathbb{Z}_{k_1}\times\cdots\times\mathbb{Z}_{k_m}\right)\cong\mathbb{Z}^{rs}\times\prod_{j=1}^{m}\mathbb{Z}_{k_j}^m\times\prod_{i=1}^{n}\prod_{j=1}^m\mathbb{Z}_{(\ell_i,k_j)}$
Of course, this gives you a theoretical way to find the Hom group between any two finitely generated abelian groups, as well.