# Class two not implies abelian automorphism group

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

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## Contents

## Statement

A group of nilpotency class two need not be a group whose automorphism group is abelian, i.e., its automorphism group need not be an abelian group.

## Related facts

- Cyclic implies abelian automorphism group
- Abelian automorphism group implies class two
- Abelian automorphism group not implies abelian

### Stronger facts

## Proof

### Example of the Klein four-group

`Further information: Klein four-group, symmetric group:S3`

Consider the Klein four-group: the direct product of two copies of the cyclic group of order two. It is an abelian group, hence it is a group of nilpotency class two.

On the other hand, its automorphism group is isomorphic to the symmetric group of degree three, which is not an abelian group.

### Example of the dihedral group of order eight

`Further information: dihedral group:D8`

The dihedral group of order eight is a non-abelian group of nilpotency class two. Its automorphism group is also isomorphic to it, i.e., it is a dihedral group of order eight. Thus, we have a group of class two whose automorphism group is not abelian.