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

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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..

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