Weakly closed implies normalizer-relatively normal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup-of-subgroup properties. That is, it states that every subgroup-of-subgroup satisfying the first subgroup-of-subgroup property (i.e., weakly closed subgroup) must also satisfy the second subgroup-of-subgroup property (i.e., normalizer-relatively normal subgroup)
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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 Normalizer-relatively normal subgroup (?) of K relative to G: in other words, H is a normal subgroup of N_G(K).

Related facts

Proof

Given: H \le K \le G such that H is weakly closed in K relative to G.

To prove: H is normal in N_G(K).

Proof: For g \in N_G(K), gHg^{-1} \le gKg^{-1} = K. Thus, gHg^{-1} \le K, so by the condition of being weakly closed, gHg^{-1} \le H. Thus, H is normal in N_G(K).