HELP:

How many homomorphisms are there from Z20 onto Z8? How many are there to Z8?

How many homomorphisms are there from Z20 onto Z8?

To map from Z_{20} \;\ to \;\ Z_{8}, you can find out where to

send the generator of Z_{20}.

Since the generator of Z_{20} has order 20, it must be mapped to an

element of order 1, 2, 4, 5, 10, or 20; these are the divisors of 20.

If you want to map them into an element of Z_{8}, then you have to map them to an element of order 1, 2, 4, or 8; these are the possible orders of elements of Z_{8}.