Cyclic automorphism group implies abelian
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
View other such results
- 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
- Group acts as automorphisms by conjugation
- Cyclicity is subgroup-closed
- Cyclic over central implies abelian
Given: A group . The automorphism group is cyclic.
To prove: is abelian.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||Let be the center of . Then, is isomorphic to a subgroup of , called the inner automorphism group .||Fact (1)||[SHOW MORE]|
|2||is a cyclic group.||Fact (2)||is cyclic||Step (1)||Step-fact-given combination direct.|
|3||is abelian||Fact (3)||Steps (1), (2)||Fact-step combination direct.|