Self-normalizing subgroup

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This is an opposite of normality

Definition

Symbol-free definition

A subgroup of a group is termed self-normalizing if it equals its own normalizer in the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed self-normalizing if $N_G(H)=H$.

Relation with other properties

Stronger properties

• Abnormal subgroup
• Weakly abnormal subgroup

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
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Let $G\le H\le K$ be groups. Then the condition that $G$ is self-normalizing in $K$ means that $N_K(G)=G$ which will imply that $N_H(G)=G$, and hence that $G$ is self-normalizing in $H$.

Thus, any self-normalizing subgroup is also self-normalizing in every intermediate subgroup.

NCI

This subgroup property is a NCI-subgroup property, i.e., it is identity-true subgroup property and further, the only normal subgroup of a group that satisfies the property is the whole group

It is clear that a subgroup that is both normal and self-normalizing must be the whole group -- that's because its normalizer equals both itself and the whole group.

Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed

An intersection of self-normalizing subgroups need not be self-normalizing. This follows from the fact that it is a NCI-subgroup property, and hence cannot be normal core-closed.