Find all homomorphisms $φ : \mathbb{Z} → \mathbb{Z}$. Which of these homomorphisms are isomorphisms?

Question

I'm currently working in the following excercise:

Find all homomorphisms $φ : \mathbb{Z} → \mathbb{Z}$. Which of these homomorphisms are isomorphisms?

If $\mathbb{Z} = \mathbb{Z}$, then would be infinite homomorphisms fulfilling the homomorphism property? I know that an isomorphism is a biyective application of $φ$ but I'm not sure about my reasoning about homomorphisms by first.

Thanks in advance for any hint or help and for taking the time to read my question.

Answer 1

The image of a map $\varphi:\mathbb{Z}\to \mathbb{Z}$ is defined by the image $\varphi(1)$, because $1$ is a generator of $\mathbb{Z}$. For example: if $\varphi(1)=3$, then $\varphi(4)=\varphi(1+1+1+1)=\varphi(1)+\varphi(1)+\varphi(1)+\varphi(1)=12$, so the map becomes multiplication with $3$. Thus we have for each $n$ a group homomorphism $\mathbb{Z}\to \mathbb{Z}$. In order to be a group isomorphism the generator $1$ has to be mapped to a generator, which can only be $1$ and $-1$, so we have $2$ isomorphisms. Does this help?