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

From Groupprops

This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group

View other such results

## Contents

## Statement

If a 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.

## Related facts

### 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

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

## Proof

**Given**: A finite nilpotent group such that every automorphism of is inner.

**To prove**: 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 is the cyclic group of order .