1. ## 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?

2. ## Re: Endomorphisms

I will assume you mean abelian group endomorphisms, and not ring endomorphisms. I will write Z4 additively.

As a group, Z4 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(Z4,Z4) has 4 elements, and in fact is isomorphic to the multiplicative monoid of Z4.

By the way, this is NOT a group, only the automorphisms Aut(Z4) form a group, of order 2. The endomorphisms: a-->0, a --> 2a, are not invertible.