What is the group of endomorphisms of $\mathbb{Q}/\mathbb{Z}$ As the question says, I'm trying to work out what $End_{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z})$ is. These are just group homomorphisms. But so far all I can see is that its probably enough to see where elements of the form $1/n$ map to, but some hints would be helpful
Thank you

By definition,
$$
\operatorname{End}_\Bbb{Z}(\Bbb{Q}/\Bbb{Z}) = \operatorname{Hom}_\Bbb{Z}(\Bbb{Q}/\Bbb{Z}, \Bbb{Q}/\Bbb{Z}),
$$
and order to define such a map, it is equivalent to think of a map
$$
f \in \operatorname{Hom}_\Bbb{Z}(\Bbb{Q}, \Bbb{Q}/\Bbb{Z})
$$
such that
$$
\Bbb{Z} \subseteq \ker f.
$$
Hint:  think about $f\left( \frac{1}{n} \right)$, for $n \in \Bbb{Z}$.
