Let $\varphi:G\to G$ . and $\varphi(g) = g^{-1}$ which I said as inverse map.

First, I proof $\varphi$ is an isomorphism by checking bijection and homomorphism.

and I know the theorem which said the inverse of isomorphic map is also isomorphic. Thus I said identity map is isomorphic.

I noted that $G$ is infinite cyclic group, so it is isomorphic to $\Bbb Z$.

So I thought the operation on $G$ should be addition "+".

Then I conclude the theorem that there is no other function that can goes from $G$ to $G$.

But I need some exact proof and want to know it's wrong or not.