This one is mostly a problem with the definitions, I think.

I am asked to prove that $\displaystyle \text{Aut} \mathbb{Z}$ is isomorphic to $\displaystyle \mathbb{Z}_2$, where $\displaystyle \text{Aut} \mathbb{Z}$ is the group of all automorphisms of $\displaystyle \mathbb{Z}$.

Obviously the identity function $\displaystyle I: \mathbb{Z} \to \mathbb{Z}: x \mapsto x$ is in $\displaystyle \text{Aut} \mathbb{Z}$. The only other automorphism I can come up with is $\displaystyle f: \mathbb{Z} \to \mathbb{Z}: x \mapsto -x$. etc, etc. to finish showing the isomorphism between $\displaystyle \text{Aut} \mathbb{Z}$ and $\displaystyle \mathbb{Z}_2$.

Are there truly only two members in $\displaystyle \text{Aut} \mathbb{Z}$? It seems there should be more, but I can't find any.

-Dan