Cyclic automorphism group implies abelian

This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group
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Statement

Suppose $G$ is a group that is an Aut-cyclic group (?): the Automorphism group (?) $\operatorname{Aut}(G)$ is a Cyclic group (?). Then, $G$ is an Abelian group (?).

Proof

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

To prove: $G$ is abelian.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Let $Z$ be the center of $G$. Then, $G/Z$ is isomorphic to a subgroup of $\operatorname{Aut}(G)$, called the inner automorphism group $\operatorname{Inn}(G)$. Fact (1) [SHOW MORE]
2 $G/Z$ is a cyclic group. Fact (2) $\operatorname{Aut}(G)$ is cyclic Step (1) Step-fact-given combination direct.
3 $G$ is abelian Fact (3) Steps (1), (2) Fact-step combination direct.