Nilpotent not implies nilpotent automorphism group

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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., nilpotent group) need not satisfy the second group property (i.e., group whose automorphism group is nilpotent)
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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 information: Klein four-group, symmetric group:S3

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