Self-normalizing subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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This is an opposite of normality

History

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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 $N_G(H)=H$.

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

This is essentially because the normalizer of a subgroup has a first-order description.

Relation with other properties

Stronger properties

• Abnormal subgroup
• Weakly abnormal subgroup

Weaker properties

• WC-subgroup
• Subgroup with canonical Abelianization: For full proof, refer: Self-normalizing implies canonical Abelianization

Incomparable properties

• Contranormal 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.

References

• Nilpotent self-normalizing subgroups of soluble groups by Roger W. Carter, Math. Zeitschr. 75, 136-139 (1961)
• Nilpotent subgroups of finite soluble groups by John S. Rose, Math. Zeitschr. 106, 97-112 (1968)

External links

Definition links

• Wikipedia page