# Nilpotent automorphism group not implies abelian automorphism group

From Groupprops

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 whose automorphism group is nilpotent) neednotsatisfy the second group property (i.e., group whose automorphism group is abelian)

View a complete list of group property non-implications | View a complete list of group property implications

Get more facts about group whose automorphism group is nilpotent|Get more facts about group whose automorphism group is abelian

## Statement

A group whose automorphism group is nilpotent (i.e., a group with nilpotent automorphism group) need not be a group whose automorphism group is abelian (i.e., a group with abelian automorphism group). In other words, the automorphism group of a group may be nilpotent and non-abelian.

## Proof

### Example of the dihedral group of order eight

`Further information: dihedral group:D8`

The dihedral group of degree four and order eight, has automorphism group isomorphic to itself. The automorphism group is thus nilpotent but not abelian.