How can I find out number of onto homomorphism from $\mathbb{Z}\to \mathbb{Z}_5$

Question

We know that, for finite groups $G,G'$ if $\phi:G\to G'$ is a onto homomorphism then, $|G'|$ divides $|G|$. But,this result will not help to predict if a homomorphism is onto(it will help to predict if a homomorphism is not onto). If I consider the homomorphism from $\mathbb{Z}\to \mathbb{Z}_5$, as $\mathbb{Z}$ is $\infty$ we can't use that result. So how I can show there are any onto homomorphism or not? and how can count total number of onto homomorphisms ?

Answer 1

Here are the key general facts:

• $\phi: \mathbb Z \to G$ is completely determined by $\phi(1)$

• $\phi: H \to \mathbb Z_n$ is surjective iff the image of $\phi$ contains a generator of $Z_n$

• the generators of $Z_n$ are $u \bmod n$ for $\gcd(u,n)=1$