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

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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

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).