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

Statement

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

Related facts

• IA-automorphism group of finite nilpotent group has precisely the same prime factors of order as the derived subgroup

Facts used

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

Proof

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