I have been given a problem where I have to find the group of all endomorphisms on $\displaystyle \mathbb{Z}_4$, $\displaystyle Hom(\mathbb{Z}_4)$. I have only been able to come up with $\displaystyle f(a) = a^n$ where $\displaystyle a\in \mathbb{Z}_4$ and $\displaystyle n$ is an integer greater than or equal to $\displaystyle 0$.

I guess the questions I'm asking are:

- Are there any more?

- How do I know if I have found them all?

- Is there a way to go about finding all the endomorphisms?