Weakly normal subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed weakly normal if whenever any conjugate of it is contained in its normalizer, the conjugate is actually contained in the subgroup itself. Equivalently, a subgroup is weakly normal if it is a weakly closed subgroup inside its normalizer relative to the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed weakly normal if for any $g$ in $G$, if $H^{g}\leq N_{G}(H)$ implies $H^{g}\leq H$.

(Here $H^{g}=g^{-1}Hg$ denotes the conjugate of $H$ by $g$ under the right action. We can use either the left or the right action for the definition).

Relation with other properties

Stronger properties

• Normal subgroup
• Pronormal subgroup: For full proof, refer: Pronormal implies weakly normal
• NE-subgroup: For proof of the implication, refer NE implies weakly normal and for proof of its strictness (i.e. the reverse implication being false) refer Weakly normal not implies NE.
• Paranormal subgroup: For proof of the implication, refer Paranormal implies weakly normal and for proof of its strictness (i.e. the reverse implication being false) refer Weakly normal not implies paranormal.
• Self-normalizing subgroup
• Weakly characteristic subgroup

Weaker properties

• Intermediately subnormal-to-normal subgroup: For full proof, refer: Weakly normal implies intermediately subnormal-to-normal
• Subgroup with self-normalizing normalizer

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
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If $H\leq K\leq G$ are groups and $H$ is weakly normal in $G$, then $H$ is weakly normal in $K$. For full proof, refer: Weak normality satisfies intermediate subgroup condition