Automorphism Group of Cyclic Group is Abelian

Theorem
Let $G$ be a cyclic group.

Let $\Aut G$ denote the automorphism group of $G$.

Then $\Aut G$ is abelian.

Proof
Let $G = \gen g$

Let $\phi, \psi \in \Aut G$.

As $G$ is cyclic:

Thus:

Thus in particular, $\phi \circ \psi$ and $\psi \circ \phi$ are equal on the generator $g$.

Since $g$ generates $G$, they must be equal as automorphisms.