Weakly closed implies conjugation-invariantly relatively normal in finite group

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Template:Subofsubgroup property implication in

Statement

Suppose H \le K \le G are groups such that H is a Weakly closed subgroup (?) of K relative to G. Then, H is a Conjugation-invariantly relatively normal subgroup (?) of K relative to G, viz., H is normal in every conjugate of K in G containing it.

Facts used

  1. Weakly closed implies normal in middle subgroup

Proof

Given: Groups H \le K \le G such that H is weakly closed in K with respect to G.

To prove: If g \in G is such that H \le gKg^{-1}, then H is normal in K.

Proof:

  1. g^{-1}Hg \le H: Since H \le gKg^{-1}, we have g^{-1}Hg \le K. Now, since H is weakly closed in K, we get that g^{-1}Hg \le H.
  2. g^{-1}Hg = H: Since G is finite, and conjugation by g is an automorphism, the sizes of g^{-1}Hg and H are the same. This, along with the previous step, yields g^{-1}Hg = H.
  3. H = gHg^{-1}: This follows from the previous step by conjugating both sides by g.
  4. H is weakly closed in gKg^{-1}: Since H is weakly closed in K, and conjugation by g is an automorphism, gHg^{-1} is weakly closed in gKg^{-1}. H = gHg^{-1} by the previos step, so H is weakly closed in gKg^{-1}.
  5. H is normal in gKg^{-1}: This follows from the previous step, and fact (1).