Finite nilpotent and every automorphism is inner implies trivial or cyclic of order two

Statement
If a fact about::finite nilpotent group is such that every automorphism of the group is inner, then the automorphism group is in fact trivial, and moreover, the group is either the trivial group or a cyclic group of order two.

Similar facts about control on automorphism group for a finite group

 * Trivial automorphism group implies trivial or order two
 * Coprime automorphism group implies cyclic with order a cyclicity-forcing number

For more results of this kind, refer:

Category:Results about control via the automorphism group

Similar and opposite facts about infinite nilpotent groups

 * There exist infinite nilpotent groups in which every automorphism is inner
 * Finitely generated nilpotent and every automorphism is inner implies trivial or cyclic of order two

Facts used

 * 1) uses::Outer automorphism group of finite nilpotent group is direct product of automorphism groups of Sylow subgroups
 * 2) uses::Group of prime power order is either trivial or of prime order or has outer automorphism class of same prime order: If $$P$$ is a finite (nontrivial) $$p$$-group for some prime $$p$$ and the order of $$P$$ is greater than $$p$$, there is an outer automorphism class of $$P$$ that has order $$p$$.

Proof
Given: A finite nilpotent group $$G$$ such that every automorphism of $$G$$ is inner.

To prove: $$G$$ is either trivial or cyclic of order two.

Proof: This follows from Facts (1) and (2), and the observation that the automorphism group of a group of order $$p$$ is the cyclic group of order $$p - 1$$.