# NE implies weakly normal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., NE-subgroup) must also satisfy the second subgroup property (i.e., weakly normal subgroup)

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## Statement

Any NE-subgroup of a group is a weakly normal subgroup.

## Definitions used

### NE-subgroup

`Further information: NE-subgroup`

A subgroup of a group is termed a NE-subgroup of if the intersection in of the normalizer and the normal closure is itself.

### Weakly normal subgroup

`Further information: Weakly normal subgroup`

A subgroup of a group is termed a weakly normal subgroup of if any conjugate of that is contained in is actually contained in .

## Proof

**Given**: A group , a subgroup such that . A conjugate of such that .

**To prove**: .

**Proof**: By definition of normal closure, . Thus, we get , so .