# Abelian automorphism group implies class two

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group whose automorphism group is abelian) must also satisfy the second group property (i.e., group of nilpotency class two)

View all group property implications | View all group property non-implications

Get more facts about group whose automorphism group is abelian|Get more facts about group of nilpotency class two

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

## Statement

Let be a group whose automorphism group is abelian, i.e., the automorphism group of is an abelian group. Then, is a group of nilpotency class two

## Related facts

### Similar facts

- Nilpotent automorphism group implies nilpotent of class at most one more
- Trivial automorphism group implies trivial or order two

- Cyclic implies abelian automorphism group
- Abelian automorphism group not implies abelian
- Locally cyclic implies abelian automorphism group
- Abelian and abelian automorphism group not implies locally cyclic
- Finite abelian and abelian automorphism group implies cyclic

## Facts used

## Proof

Since is an abelian group, the subgroup , the inner automorphism group is also an abelian group. But , where is the center of . Thus, is abelian, so has class two.