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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.

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    What kind of homomorphisms? Group homomorphisms and ring homomorphisms, for example, satisfy different conditions, so there are maps that are one but not the other.2011-12-28
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    @Junior: Presumably you meant $\mathbb{Z}_2$, not $\mathbb{Z}^2$, since you used the word "onto". Am I correct?2011-12-28
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    A homomorphism $\mu: \mathbb{Z} \to G$ (where $G$ is a group) is uniquely determined by $\mu(1)$ (can you see why?).2011-12-28
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    @Henning Makholm:group homomorphisms2011-12-28
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    @Zev:yes correct2011-12-28
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    @Pedro:i don't know why2011-12-28
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    @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

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