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.

Related facts

Similar facts

• Trivial automorphism group implies trivial or cyclic of order two
• Finite abelian and aut-abelian implies cyclic
• Cyclic implies aut-abelian
• Aut-cyclic implies abelian
• Aut-abelian implies class at most two

Opposite facts

• Aut-cyclic not implies cyclic
• Aut-abelian not implies abelian
• Abelian and aut-abelian not implies locally cyclic

Facts used

1. Aut-cyclic implies abelian
2. Cyclic implies abelian
3. Finite abelian and aut-abelian implies 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.