Let's make an arbitrary choice for .
Now what does have to be (and why)?
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?
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 , ).