Proving that there aren't any more homomorphisms

I was asked to find all homomorphisms

I'm fairly convinced in my own mind that all of them are

I can show that these are homomorphisms by claiming that is defined as above and:

1. (definition of homomorphism)

2. (1, my def. of )

3. (2, distributive prop.)

So, how do I show now that there are no other homomorphisms?

Re: Proving that there aren't any more homomorphisms

Hi beebe! :)

Let's make an arbitrary choice for .

Suppose .

Now what does have to be (and why)?

Re: Proving that there aren't any more homomorphisms

more generally, if G is any cyclic group, with generator x, and φ:G→G is a homomorphism, you can show that φ is completely determined by φ(x).

(because ). so if , we have .

now is cyclic, with generator 1, so....

(recall that in , ).