# Weakly closed implies normalizer-relatively normal

From Groupprops

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 are groups such that is a Weakly closed subgroup (?) of relative to . Then, is a Normalizer-relatively normal subgroup (?) of relative to : in other words, is a normal subgroup of .

## Related facts

## Proof

**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 .