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 H of a group G is termed a NE-subgroup of G if the intersection in G of the normalizer N_G(H) and the normal closure H^G is H itself.

Weakly normal subgroup

Further information: Weakly normal subgroup

A subgroup H of a group G is termed a weakly normal subgroup of G if any conjugate of H that is contained in N_G(H) is actually contained in H.

Proof

Given: A group G, a subgroup H such that H = H^G \cap N_G(H). A conjugate H^g of H such that H^g \le N_G(H).

To prove: H^g \le H.

Proof: By definition of normal closure, H^g \le H^G. Thus, we get H^g \le H^G \cap N_G(H) = H, so H^G \le H.