Cyclic implies abelian automorphism group

Statement
Any cyclic group is a group whose automorphism group is abelian: the automorphism group of a cyclic group is an abelian group.

Related facts

 * Aut-abelian not implies abelian

Proof
Given: A cyclic group $$G$$ with generator $$g$$. Automorphisms $$\alpha, \beta$$ of $$G$$.

To prove: $$\alpha \circ \beta = \beta \circ \alpha$$.

Proof: Suppose $$\alpha, \beta$$ are two automorphisms of $$G$$. Since $$G$$ is cyclic on $$g$$, there exist integers $$a,b$$ such that $$\alpha(g) = g^a, \beta(g) = g^b$$. Thus, we have:

$$\alpha \circ \beta (g) = \beta \circ \alpha (g) = g^{ab}$$.

In particular, $$\alpha \circ \beta$$ and $$\beta \circ \alpha$$ are equal on the generator $$g$$. Since $$g$$ generates $$G$$, they must be equal as automorphisms.