Finite and cyclic automorphism group implies cyclic

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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

Opposite facts

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.

Hands-on proof