# IA-automorphism group of finite p-group is p-group

From Groupprops

## Contents

## Statement

Suppose is a prime number and is a finite p-group. Then, the group of IA-automorphisms of is also a finite -group (for the same prime ). In particular, every IA-automorphism of has order a power of .

## Related facts

## Facts used

- Prime power order implies nilpotent
- IA-automorphism group of nilpotent group equals stability group of lower central series
- Stability group of subnormal series of finite p-group is p-group

## Proof

The proof follows directly by combining Facts (1), (2), and (3).