
Endomorphisms
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?

Re: Endomorphisms
I will assume you mean abelian group endomorphisms, and not ring endomorphisms. I will write Z_{4} additively.
As a group, Z_{4} is generated by 1, so any endomorphism is completely determined by the image of 1. We have 4 choices for such an endomorphism f:
f(1) = 0. This is the trivial map, which sends everything to 0.
f(1) = 1. This is the identity map, which is an automorphism.
f(1) = 2. This is a quotient map onto the subgroup {0,2} (which sends a to 2a (mod 4)).
f(1) = 3. This is another automorphism, which sends a to a.
So Hom(Z_{4},Z_{4}) has 4 elements, and in fact is isomorphic to the multiplicative monoid of Z_{4}.
By the way, this is NOT a group, only the automorphisms Aut(Z_{4}) form a group, of order 2. The endomorphisms: a>0, a > 2a, are not invertible.