Cyclic automorphism group implies abelian
From Groupprops
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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Contents
Statement
Suppose is a group that is an Aut-cyclic group (?): the Automorphism group (?)
is a Cyclic group (?). Then,
is an Abelian group (?).
Related facts
- Finite and cyclic automorphism group implies cyclic
- Classification of finite groups with cyclic automorphism group
- Cyclic automorphism group not implies cyclic
- Finite abelian and abelian automorphism group implies cyclic
Facts used
- Group acts as automorphisms by conjugation
- Cyclicity is subgroup-closed
- Cyclic over central implies abelian
Proof
Given: A group . The automorphism group
is cyclic.
To prove: is abelian.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Let ![]() ![]() ![]() ![]() ![]() |
Fact (1) | [SHOW MORE] | ||
2 | ![]() |
Fact (2) | ![]() |
Step (1) | Step-fact-given combination direct. |
3 | ![]() |
Fact (3) | Steps (1), (2) | Fact-step combination direct. |