Cyclic automorphism group implies abelian

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

Related facts

Facts used

  1. Group acts as automorphisms by conjugation
  2. Cyclicity is subgroup-closed
  3. Cyclic over central implies abelian

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.