Characterizing the collection of automorphisms on $\mathbb{Z}$ with a binary relation.

Question

How can one characterize the collection of automorphisms of integers $\mathbb{Z}$ with the binary relation "$<$"? Or "$>$"? "$=$"? How can we acquire the collection of automorphisms?

Answer 1

HINT: Suppose that $f$ is an automorphism, and $f(0)=n$, what can you tell about $f(1)$ and about $f(-1)$? Show that $f(0)$ decides completely all the values of $f$. Now find a compact and nice way to write $f$.

Answer 2

$\Bbb Z=<1>$ or $<-1>$. So all the automorphisms can be created by sending $1$ to $1$ or $-1$. So, there are two automorphisms. So, $Aut(\Bbb Z) \equiv \Bbb Z_2=<-1>$.