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

From Groupprops

## Statement

Suppose is a periodic nilpotent group that is also a group in which every automorphism is inner. Then, must be either a trivial group or cyclic group:Z2.

## Facts used

- Nilpotent p-group and every automorphism is inner implies trivial or cyclic or order two
- Outer automorphism group of periodic nilpotent group is direct product of outer automorphism groups of Sylow subgroups

## Proof

The proof follows directly from Facts (1) and (2).