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Class-preserving automorphism group of finite p-group is p-group
From Groupprops
Statement
Suppose P is a finite p-group, i.e., a group of prime power order where the prime is p. Let 
denote the group of class-preserving automorphisms of P, i.e., the automorphisms that send every element to within its conjugacy class. Then, is also a p-group.
Facts used
- Class-preserving implies stability automorphism of central series
- Stability group of subnormal series of p-group is p-group
Proof
The proof follows directly from facts (1) and (2).
