TITLE: How many group homomorphisms we can get from ${\mathbb Z}_{20}$ to ${\mathbb Z}_{10}$?
QUESTION [5 upvotes]: How many group homomorphisms we can get from ${\mathbb Z}_{20}$  to   $ {\mathbb Z}_{10}$?
My Try: I think $4$. Because $1 , 2 , 5 , 10$ are the only possible order of image of ${\mathbb Z}_{20} $. 
Can anyone please help me?

REPLY [5 votes]: Any homomorphism is completely determined by the image of $1$. That is, if $1 \mapsto a$, then $x \mapsto xa$.
So assume $1 \mapsto a$. Then by Lagrange's theorem $\vert a \vert$ divides $10$. Also by this result$\Bigl( \text{If}\; \vert g \vert$ is finite, then $\vert \phi(g) \vert$ divides $\vert g \vert \Bigr)$,we have $\vert a \vert$ divides $20$ ,since $\vert 1 \vert =20$ is finite. 
Thus $\vert a \vert$ divides both $20$ and $10$. So $\vert a \vert=1,2,5,10$
Thus $a=0,5,2,8,4,6,1,3,7,9$. ( Actually the range of this "$a$" is all the elements of $\Bbb{Z}_{10} $, but I am writing in the order mentioned above). This give a list of $10$ 
homomorphisms.
That is
$$x \mapsto ax$$ where $a=0,1,2,...,9$ are all the required ten homomorphisms.
In General, the number of group homomorphisms from $\Bbb{Z}_m$ to $\Bbb{Z}_n$ is $\text{gcd}(m,n)$