Self-normalizing subgroup

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Names in other languages:British English: self-normalising subgroup

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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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RANDOM TIP:The metaproperties section lists important facts about the subgroup property, and addresses many of the natural questions that arise about it. It has links to proofs.

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.

In terms of the in-normalizer operator

This property is obtained by applying the in-normalizer operator to the property: improper subgroup
View all properties obtained by applying the in-normalizer operator

Relation with other properties

Stronger properties

Abnormal subgroup
Weakly abnormal subgroup
Free factor: Any nontrivial free factor of a group is either self-normalizing or trivial. For full proof, refer: Free factor implies self-normalizing or trivial

Weaker properties

WC-subgroup
Subgroup with canonical Abelianization: For full proof, refer: Self-normalizing implies canonical Abelianization
Self-centralizing subgroup: For proof of the implication, refer Self-normalizing implies self-centralizing and for proof of its strictness (i.e. the reverse implication being false) refer Self-centralizing not implies self-normalizing.

Incomparable properties

Contranormal subgroup: For full proof, refer: Self-normalizing not implies contranormal, Contranormal not implies self-normalizing

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
View a complete list of transitive subgroup properties|View facts related to transitivity of subgroup properties

A self-normalizing subgroup of a self-normalizing subgroup need not be self-normalizing.

Further information: Self-normalizing is not transitive

Intermediate subgroup condition

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the 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, viz 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.

Join-closedness

This subgroup property is not join-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 join-closed

A join of self-normalizing subgroups need not be self-normalizing. This follows because the property of being self-normalizing is not normal closure-closed: there exist self-normalizing subgroups whose normal closure is a proper normal subgroup.

Image condition

This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View a complete list of subgroup properties satisfying the image condition

If $H$ is a self-normalizing subgroup of $G$ , and $f:G \to K$ is a surjective homomorphism of groups, then $f (H)$ is a self-normalizing subgroup of $K$ .

Further information: Self-normalizing satisfies image condition

Direct product-closedness

This subgroup property is direct product-closed: it is closed under taking arbitrary direct products of groups

If $H 1$ is a self-normalizing subgroup of math>G_1</math>, and $H 2$ is a self-normalizing subgroup of $G 2$ , then $H_1 \times H_2$ is a self-normalizing subgroup of $G_1 \times G_2$ . The analogous statement holds for arbitrary direct products as well.

For full proof, refer: Self-normalizing is direct product-closed

Effect of property operators

The upward-closure

Applying the upward-closure to this property gives: weakly abnormal subgroup

If $H$ is a subgroup of $G$ such that every subgroup of $G$ containing $H$ is self-normalizing in $G$ , then $H$ is termed a weakly abnormal subgroup of $G$ . Being weakly abnormal is also equivalent to being contranormal in every intermediate subgroup.

Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsSelfNormalizing
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

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

Applying operator gives	Weakly abnormal subgroup +
Defining ingredient	In-normalizer operator +, and Improper subgroup +
Referenced in	Wikipedia (?, ?, ?) +
Stronger than	WC-subgroup +, Subgroup with canonical Abelianization +, and Self-centralizing subgroup +
Weaker than	Abnormal subgroup +, Weakly abnormal subgroup +, and Free factor +