Finite and cyclic automorphism group implies cyclic

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 finite group that is an Aut-cyclic group (?), i.e., the automorphism group of $G$ is a cyclic group. Then, $G$ is itself a Cyclic group (?), and in particular, a Finite cyclic group (?).

Note that the converse is not true, i.e., the automorphism group of a finite cyclic group need not be cyclic.

Proof

Proof from given facts

Given: A finite group $G$ such that $\operatorname{Aut}(G)$ is cyclic.

To prove: $G$ is cyclic.

Proof

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 $G$ is abelian. Fact (1) $\operatorname{Aut}(G)$ is cyclic. -- given + fact direct
2 $\operatorname{Aut}(G)$ is abelian. Fact (2) $\operatorname{Aut}(G)$ is cyclic. -- given + fact direct
3 $G$ is cyclic Fact (3) $G$ is finite Steps (1), (2) Fact + step + given direct.