View source for Nilpotent not implies nilpotent automorphism group

{{group property non-implication|
stronger = nilpotent group|
weaker = group whose automorphism group is nilpotent}}

==Statement==

The [[automorphism group]] of a [[nilpotent group]] need not be nilpotent. In other words, a nilpotent group is not necessarily an [[group whose automorphism group is nilpotent]].

==Proof==

===Example of the Klein four-group===

{{further|[[particular example::Klein four-group]], [[particular example::symmetric group:S3]]}}

The [[automorphism group]] of the [[Klein four-group]], which is an abelian and hence nilpotent group, is the [[symmetric group:S3|symmetric group of degree three]], which is not a nilpotent group.