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) uses::Prime power order implies nilpotent
 * 2) uses::IA-automorphism group of nilpotent group equals stability group of lower central series
 * 3) uses::Stability group of subnormal series of finite p-group is p-group

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