Class-preserving automorphism group of finite p-group is p-group

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Statement

Suppose P is a finite p-group, i.e., a Group of prime power order (?) where the prime is p. Let \operatorname{Class}(P) denote the group of Class-preserving automorphism (?)s of P, i.e., the automorphisms that send every element to within its conjugacy class. Then, \operatorname{Class}(P) is also a p-group.

Facts used

  1. Class-preserving implies stability automorphism of central series
  2. Stability group of subnormal series of p-group is p-group

Proof

The proof follows directly from facts (1) and (2).