Number of homomorphisms between groups

Question

I have difficulty solving problems involving homomorphisms. There are some problems in my textbook that suggest problems as follows:

How many homomorphisms are there from $\Bbb Z_{20}$ to $\Bbb Z_{8}?$

or

Determine all the homomorphisms from $\Bbb Z_4$ to $\Bbb Z_2\oplus \Bbb Z_2$

Can someone please give me a hint or a sketch of how to do these kinds of problems?

Thanks in advance!

Answer 1

For the first problem, any group homomorphism $\phi:\mathbb{Z}/20\mathbb{Z}\to \mathbb{Z}/8\mathbb{Z}$ is determined by $\phi(1)$, which must satisfy $20\phi(1)=0$. So how many choices are there for $\phi(1)$?

Answer 2

Show that any group homomorphism between $\mathbb{Z}_{n}$ and $\mathbb{Z}_{m}$ are determined by $\phi(x)=\phi(1)\overline {x}$, where the order of $\phi(1)$ divides $d=gcd(n,m)$, and there is going to be $d$ homomorphism between $\mathbb{Z}_{n}$ and $\mathbb{Z}_{m}$.