Isomorph-normal characteristic of WNSCDIN implies weakly closed

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Statement

Suppose H \le K \le G are groups. Suppose, further, that H is an Isomorph-normal characteristic subgroup (?) of K and K is a WNSCDIN-subgroup (?) of G. Then, H is a Weakly closed subgroup (?) of K relative to G.

Facts used

  1. Characteristic of normal implies normal
  2. WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Proof

Given: Groups H \le K \le G, such that H is characteristic in K, every subgroup of K isomorphic to H is normal in K, and K is a WNSCDIN-subgroup of G.

To prove: H is weakly closed in G.

Proof:

  1. H is normal in N_G(K): Since H is characteristic in K and K is normal in N_G(K), fact (1) yields that H is normal in N_G(K).
  2. H is normal in every conjugate of K containing it: Suppose H \le gKg^{-1} for some g \in G. Then, g^{-1}Hg \le K. Clearly, g^{-1}Hg is isomorphic to H. So, by the assumption, g^{-1}Hg is normal in K. Conjugating back, we get that H is normal in gKg^{-1}.
  3. H is weakly closed in K with respect to G: This follows from fact (2), using the previous two steps.