Odd-order p-group has coprime automorphism-faithful characteristic class two subgroup of prime exponent

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This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a finite group being coprime automorphism-faithful. In other words, any non-identity automorphism of of the whole group, of coprime order to the whole group, that restricts to the subgroup, restricts to a non-identity automorphism of the subgroup.
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Suppose p is an odd prime and P is a finite p-group (i.e., a group of prime power order). Then, there exists a subgroup K of P satisfying the following conditions:

Facts used

  1. Thompson's critical subgroup theorem
  2. Omega-1 of odd-order p-group is coprime automorphism-faithful
  3. Omega-1 of odd-order class two p-group has prime exponent
  4. Coprime automorphism-faithful characteristicity is transitive
  5. Nilpotence class two is subgroup-closed