Cyclic automorphism group implies abelian

Statement
Suppose $$G$$ is a group that is an fact about::aut-cyclic group: the fact about::automorphism group $$\operatorname{Aut}(G)$$ is a fact about::cyclic group. Then, $$G$$ is an fact about::abelian group.

Related facts

 * Finite and cyclic automorphism group implies cyclic
 * Classification of finite groups with cyclic automorphism group
 * Cyclic automorphism group not implies cyclic
 * Finite abelian and abelian automorphism group implies cyclic

Facts used

 * 1) uses::Group acts as automorphisms by conjugation
 * 2) uses::Cyclicity is subgroup-closed
 * 3) uses::Cyclic over central implies abelian

Proof
Given: A group $$G$$. The automorphism group $$\operatorname{Aut}(G)$$ is cyclic.

To prove: $$G$$ is abelian.

Proof: