Self-normalizing subgroup
Names in other languages:British English: self-normalising subgroupUse Google translate to translate this page to French, German, Spanish, Italian
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, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is an opposite of normality
Contents
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 of a group is termed self-normalizing if .
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 other 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
- Central factor of normalizer
- 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
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity
A self-normalizing subgroup of a self-normalizing subgroup need not be self-normalizing.
Further information: Self-normalizing is not transitive
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 be groups. Then the condition that is self-normalizing in means that which will imply that , and hence that is self-normalizing in .
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.
Join-closedness
This subgroup property is not join-closed, viz., it is not true that a join 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
YES: 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 other subgroup properties satisfying image condition
If is a self-normalizing subgroup of , and is a surjective homomorphism of groups, then is a self-normalizing subgroup of .
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 is a self-normalizing subgroup of math>G_1</math>, and is a self-normalizing subgroup of , then is a self-normalizing subgroup of . 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 is a subgroup of such that every subgroup of containing is self-normalizing in , then is termed a weakly abnormal subgroup of . 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.GAP-codable subgroup property
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)