TITLE: Why does the set of all automorphisms on the additive group of integers contain only 2 elements?
QUESTION [6 upvotes]: I believe that $f(x)=x$ and $f(x)=-x$ are the only two automorphisms.
But the set of automorphisms on $\mathbb{Z}_n$ is isomorphic to $U_n$.
So as $n$ increases, the number of automorphisms on $\mathbb{Z}_n$ increases.
However, for the entire set of integers, there are only two automorphisms.
I want to understand why this changes at infinity.
Why isn't there a limiting property?

REPLY [3 votes]: All these groups are 1-generated.  Thus, the image of a generator must be a generator. (Isomorphisms preserve such things.) But
$\Bbb Z$ has only two generators, whereas $\Bbb Z_n$ can have many.  In fact, the number of automorphisms of $\Bbb Z_n$ equals the number of its generators.