Weakly normal not implies NE

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., weakly normal subgroup) need not satisfy the second subgroup property (i.e., NE-subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about weakly normal subgroup|Get more facts about NE-subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property weakly normal subgroup but not NE-subgroup|View examples of subgroups satisfying property weakly normal subgroup and NE-subgroup

Statement

A weakly normal subgroup of a group need not be a NE-subgroup.

Related facts

• NE implies weakly normal

Facts used

1. Pronormal implies weakly normal
2. Pronormal not implies NE

Proof

The proof follows directly from facts (1) and (2). More specifically, any example of fact (2) also gives an example of a weakly normal subgroup that is not NE.