How can I find the $\mathrm{End}_{\mathbb Z_{20}}(5\mathbb Z_{20})$

Question

I know that it is isomorphic to $\mathbb Z_4$. but I cant prove it. Any comment will be appreciated.

Answer 1

Hints:

• the $\mathbb{Z}_n$- and $\mathbb{Z}$-module endomorphisms of a $\mathbb{Z}_n$-module are the same,
• $d\mathbb{Z}_n\cong\mathbb{Z}_{n/d}$ as abelian groups when $d\mid n$,
• any homomorphism out of $\mathbb{Z}_m$ is determined by where it sends a generator