How many homomorphisms are there from Z20 onto Z10? How many are there to Z10?
All homomorphisms coming from $\displaystyle \mathbb{Z}_{20}$ are entirely determined by where they send 1. $\displaystyle \phi(n)=\phi(1+1+...+1)=\phi(1)+\phi(1) +... + \phi(1) = n\phi(1)$
If it is to be onto, it better send 1 to a generator of $\displaystyle \mathbb{Z}_{10}$ but this is exactly the integers that are relatively prime to 10. $\displaystyle U(10)=\{1,3,7,9\}$ so I count 4 distinct homomorphisms that will take $\displaystyle \mathbb{Z}_{20}$ onto $\displaystyle \mathbb{Z}_{10}$
I think you just gotta check that the order of the image divides the order of the preimage. I think actually in this case all of them are okay to go to since by lagrange, the order of the image must divide 10 and 10 divides 20 which is the order of 1, so as far as I can tell there should be 10 possibilities for just regular homomorphisms.
you should check though, I mean it is easy to do just check if $\displaystyle \phi(a+b)=\phi(a)+\phi(b)$ for the other 6 cases or as many as you need to to convince yourself it works.
Can we generalize how many homomorphisms there will be onto and into a group depending on the size of the group (particularly if we are working with Z groups)? For example, if we are looking for the number of homomorphisms from Z20 onto and into Z8, what would we find?
Yeah man just look at how I did the onto part for your original question. Just for simplicity sake lets talk about homomorphisms from $\displaystyle \mathbb{Z}_m$ onto $\displaystyle \mathbb{Z}_n$
Clearly if $\displaystyle m < n$ there are 0 because there are not enough things in $\displaystyle \mathbb{Z}_m$ to hit everything in $\displaystyle \mathbb{Z}_n$ just by counting.
But in the other cases as I said before any $\displaystyle \phi$ is completely determined by where it sends 1. $\displaystyle \phi(n)=n\phi(1)$. So if this is going to be onto $\displaystyle \phi(1)$ MUST be a generator of $\displaystyle \mathbb{Z}_n$. These are exactly the numbers $\displaystyle 0 \leq a \leq n-1$ such that $\displaystyle (a,n)=1$ the numbers relatively prime to n; the set of units of $\displaystyle \mathbb{Z}_n$
Look up Euler's totient function to see explicitly how many this is for any given number. Euler's totient function - Wikipedia, the free encyclopedia
For 8 just count them. how many numbers less than 8 share no divisors other than 1?