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

Statement
Suppose $$G$$ is a finitely generated nilpotent group (i.e., it is both a finitely generated group and a nilpotent group) that is also a  group in which every automorphism is inner. Then, $$G$$ is either the trivial group or is isomorphic to cyclic group:Z2.

Another way of putting this is that any nilpotent group in which every automorphism is inner must be either the trivial group, or cyclic group:Z2, or an infinite (and non-finitely generated) nilpotent group..

Similar facts

 * Finite nilpotent and every automorphism is inner implies trivial or cyclic of order two
 * Group of prime power order is either trivial or of prime order or has outer automorphism class of same prime order

Opposite facts

 * There exist infinite nilpotent groups in which every automorphism is inner