Automorphisms of the group of integers $\mathbb Z$ Can anyone help me showing $\operatorname{Aut}(\mathbb Z)\simeq \mathbb Z_2$? 
I guess I should define an homomorfism $\phi:\mathbb Z\longrightarrow S(\mathbb Z)$ with kernel $2\mathbb Z$ and image $\operatorname{Aut}(\mathbb Z)$. Here $S(\mathbb Z)$ are the bijections of $\mathbb Z$. However I wasn't able to do that.
Any help will be welcome.

Let $\varphi \in \operatorname{Aut}(\mathbb Z)$. We have
$$
\varphi(n) = \varphi(\underbrace{1 + \cdots + 1}_{n \text{ times}}) = \underbrace{\varphi(1) + \cdots + \varphi(1)}_{n \text{ times}} = n \ \varphi(1).
$$
Thus, $\varphi$ is completely determined by its value at $1$. Only two values make $\varphi$ surjective, $1$ and $-1$. It follows that $\operatorname{Aut}(\mathbb Z)$ consists of two elements. In other words, $\operatorname{Aut}(\mathbb Z) \cong \mathbb Z / 2\mathbb Z$.
