Determine all homomorphism $\varphi : \mathbb{Z}^{+}\rightarrow \mathbb{Z}^{+}$

Question

Determine all homomorphism $\varphi : \mathbb{Z}^{+}\rightarrow \mathbb{Z}^{+}$, and determine which are injective, surjective, and isomorphism.

Answer of first part:Note that $\varphi (a+b) =\varphi(a)+\varphi(b), a,b\in \mathbb{Z}^+$

Then for $n\in \mathbb{Z}^+$

$\varphi (n) =\varphi (1+1+\dots + 1\space[\text{$n$ times]}), \text{when}\space n>0\\ \varphi(n)=\varphi(0) ,\space \text{when $n=0$}\\ \varphi (n) =\varphi (-1-1-\dots - 1\space[\text{$n$ times]}), \text{when}\space n<0$

i.e.$ \varphi (n) =n\varphi(1)$

Surjective part is clear since it is only for $\varphi(1)=+1, -1$.

Now if $\varphi(1)\neq 0$ all $\varphi$'s are injective, isn't??And what about isomorphism? Where am I wrong!! Please help me.

Answer 1

All $\varphi$ are injective (except the trivial homomorphism) as $|Ker (\varphi)|=1$. The only surjective homomorphism is $\varphi(1)=1,-1$, else there is no $n$ such that $\varphi(n)=1$. This also determines the only isomorphisms.