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Groupprops β

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

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

Proof

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