Weakly closed implies normalizer-relatively normal
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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- WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed
Given: such that is weakly closed in relative to .
To prove: is normal in .
Proof: For , . Thus, , so by the condition of being weakly closed, . Thus, is normal in .