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Let $\mathbb{Q}$ be the group ($\mathbb{Q}$,+) and $\mathbb{Z}$ is a sub-group of $\mathbb{Q}$.

It is quite easy to find all homomorphism from $\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Q}/ \mathbb{Z}$.

However, I couldn’t find what would be all homomorphism from $\mathbb{Q}/ \mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$.

Please help me.

- What is the field of fractions of $\mathbb{Q}/(x^2+y^2)$?
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- $\mathbb{Q}/\mathbb{Z}$ is divisible
- Theorems implied by Yoneda's lemma?

Because there is only the trivial one!

Any morphism $\Bbb Q/\Bbb Z\to\Bbb Z/n\Bbb Z$ is induced by a morphism $\phi:\Bbb Q\to \Bbb Z/n\Bbb Z$ such that $\phi(1)=0$.

Now let $u\in\Bbb Q$ arbitrary, then we must have

$$\phi(u)=\phi\left(n\cdot\frac un\right)=n\cdot \phi\left(\frac un\right)=0$$

in $\Bbb Z/n\Bbb Z$.

Another approach:

As a homomorphic image of a divisible group, the quotient $\;\Bbb Q/\Bbb Z\;$ is divisible, and thus the image of any homomorphism from this group is also divisible. But the only divisible **finite** group is the trivial one, and thus the only possible homomorphism $\;\Bbb Q/\Bbb Z\to\Bbb Z/n\Bbb Z\;$ is the trivial one.

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