Isomorph-normal coprime automorphism-invariant of Sylow implies weakly closed

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a group of prime power order. That is, it states that in a Group of prime power order (?), every subgroup satisfying the first subgroup property (i.e., Isomorph-normal coprime automorphism-invariant subgroup (?)) must also satisfy the second subgroup property (i.e., Sylow-relatively weakly closed subgroup (?)). In other words, every isomorph-normal coprime automorphism-invariant subgroup of group of prime power order is a Sylow-relatively weakly closed subgroup of group of prime power order.
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Statement

Suppose P is a group of prime power order (where the underlying prime is p) and H is an isomorph-normal coprime automorphism-invariant subgroup of P. In other words, we have the following:

  • H is isomorph-normal in P: Any subgroup of P isomorphic to H is normal in P.
  • H is coprime automorphism-invariant in P: Any p'-automorphism of P leaves H invariant.

Then, for any finite group G containing P as a p-Sylow subgroup, H is weakly closed in P with respect to G.

Related facts

Facts used

  1. Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal
  2. Isomorph-normal implies conjugation-invariantly relatively normal in any ambient group
  3. Sylow implies WNSCDIN
  4. WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Proof

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