Pronormal subgroup
Definition
QUICK PHRASES: conjugates in whole group are conjugate in intermediate subgroups, conjugates in whole group are conjugate in join
Symbol-free definition
A subgroup of a group is termed pronormal if it satisfies the following equivalent conditions:
No. | Shorthand | A subgroup of a group is pronormal if ... | A subgroup of a group is pronormal if... (right-action convention) | A subgroup of a group is pronormal if... (left-action convention) |
---|---|---|---|---|
1 | Conjugates in whole group are conjugate in intermediate subgroups | any conjugate subgroup of the subgroup inside the whole group is also conjugate inside any intermediate subgroup. | for any and any subgroup of containing both and , there exists such that . | for any and any subgroup of containing both and , there exists such that . |
2 | Conjugates in whole group are conjugate in join | any conjugate subgroup of the subgroup is conjugate to it inside the subgroup generated by the original subgroup and its conjugate. | for any in , there exists such that . Here and the angled braces are for the join of subgroups (or subgroup generated). | for any , there exists such that . |
Under the right action convention, the right action of on by conjugation gives the subgroup . Under the left action convention, the left action of on by conjugation gives the subgroup . Note that although the actions differ, the notion of being conjugate subgroups inside an intermediate subgroup remains unchanged. This can be explained by the fact that the inverse map reverses the roles of left and right while preserving subgroups.
Equivalence of definitions
The two definitions are equivalent because being conjugate inside the smallest possible intermediate subgroup, viz , implies being conjugate in any intermediate subgroup.
This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
VIEW: Definitions built on this | Facts about this: (facts closely related to Pronormal subgroup, all facts related to Pronormal subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions
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 a variation of normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup
History
Origin
This term was introduced by: Hall
The notion of pronormal subgroup was introduced by Philip Hall and the first nontrivial results on it were obtained by John S. Rose in his paper Finite soluble groups with pronormal system normalizers.
Examples
VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Extreme examples
- Every group is pronormal as a subgroup of itself
- The trivial subgroup is always pronormal.
Generic examples
- All Sylow subgroups are pronormal.
- Maximal subgroups and normal subgroups are pronormal.
Particular examples
- High occurence example: In the symmetric group of degree three, all subgroups are pronormal.
- Low occurrence example: In a nilpotent group, the pronormal subgroups are the same as the normal subgroups. That's because every subgroup is subnormal, and pronormal and subnormal implies normal.
Non-examples
- In a symmetric group of order four, or in a symmetric group of higher order, a subgroup generated by a transposition is not pronormal. That's because conjugating it can give a subgroup generated by a disjoint transposition.
- A subnormal subgroup that is not normal, cannot be pronormal. That's because pronormal and subnormal implies normal.
Formalisms
Monadic second-order description
This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups
View other monadic second-order subgroup properties
Pronormality can be expressed using a monadic second-order sentence. The sentence is somewhat complicated. First, note that, using monadic second-order logic, it is possible to construct the subgroup generated by any subset (namely as the smallest subset containing that subset and closed under group operations). Thus, if is a subgroup of , the group can be constructed using monadic second-order logic. Pronormality testing is now the following sentence:
Relation with other properties
Related survey articles
- Subnormal-to-normal and normal-to-characteristic: This survey article discusses in detail a number of subgroup properties that, along with subnormality, imply normality. Pronormality is a prominent one among them. The article also has this implication diagram that shows the implication relations between various subgroup properties.
Stronger properties
For a complete list of subgroup properties stronger than Pronormal subgroup, click here
STRONGER PROPERTIES SATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
STRONGER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
Weaker properties
- Subnormal-to-normal subgroup: For full proof, refer: Pronormal and subnormal implies normal
- Intermediately subnormal-to-normal subgroup: For full proof, refer: Pronormal implies intermediately subnormal-to-normal. For intermediate notions between intermediately subnormal-to-normal subgroup and pronormal subgroup, click here. Also related:
- Weakly pronormal subgroup: For proof of the implication, refer Pronormal implies weakly pronormal and for proof of its strictness (i.e. the reverse implication being false) refer Weakly pronormal not implies pronormal.
- Paranormal subgroup: For proof of the implication, refer Pronormal implies paranormal and for proof of its strictness (i.e. the reverse implication being false) refer Paranormal not implies pronormal.
- Polynormal subgroup
- Weakly normal subgroup
For a survey article exploring these properties in greater depth, refer: subnormal-to-normal and normal-to-characteristic
- Self-conjugate-permutable subgroup: For proof of the implication, refer Pronormal implies self-conjugate-permutable and for proof of its strictness (i.e. the reverse implication being false) refer Self-conjugate-permutable not implies pronormal.
- Join of pronormal subgroups
- Subgroup with abnormal normalizer: For full proof, refer: Normalizer of pronormal implies abnormal Also related:
- WNSCDIN-subgroup: For proof of the implication, refer Pronormal implies WNSCDIN and for proof of its strictness (i.e. the reverse implication being false) refer WNSCDIN not implies pronormal. Also related:
- MWNSCDIN-subgroup: For proof of the implication, refer Pronormal implies MWNSCDIN and for proof of its strictness (i.e. the reverse implication being false) refer MWNSCDIN not implies pronormal.
- Subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed: For full proof, refer: WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed
- Subgroup having a pronormalizer
- Subgroup having a subnormalizer
View a more comprehensive list of subgroup properties weaker than pronormality
Conjunction with other properties
- Any pronormal subnormal subgroup is normal, and conversely, a normal subgroup is both pronormal and subnormal: For full proof, refer: Pronormal and subnormal implies normal, Normal implies pronormal, Normal implies subnormal, Pronormal implies intermediately subnormal-to-normal
- A subgroup is intermediately isomorph-conjugate if and only if it is both pronormal and isomorph-conjugate. For full proof, refer: Pronormal and isomorph-conjugate equals intermediately isomorph-conjugate
Related group properties
- Group in which every subgroup is pronormal: Such groups are, in particular, T*-groups, and T-groups: normality is transitive on subgroups of such groups.
- Group in which every weakly pronormal subgroup is pronormal: All solvable groups, and more generally, all hyper-N-groups, satisfy this condition.
- Group in which every pronormal subgroup is normal: This property is satisfied by all locally nilpotent groups. For finite groups, it is equivalent to being a finite nilpotent group.
Metaproperties
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)
Trimness
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
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
Pronormality satisfies the intermediate subgroup condition, that is, any pronormal subgroup is pronormal in every intermediate subgroup. Further information: Pronormality satisfies intermediate subgroup condition
Transfer condition
This subgroup property does not satisfy the transfer condition
If is pronormal and , then need not be pronormal in . Further information: Pronormality does not satisfy transfer condition
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 pronormal subgroups need not be pronormal. In fact, even a finite intersection of pronormal subgroups need not be pronormal. For full proof, refer: Pronormality is not finite-intersection-closed
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
The property of pronormality is not transitive. This follows because every normal subgroup is pronormal and every pronormal subnormal subgroup is normal. The proof generalizes to all properties sandwiched between normality and the property of being subnormal-to-normal.
The subordination of this property is the property of being subpronormal.
Further information: Pronormality is not transitive, Subgroup property between normal and subnormal-to-normal is not transitive, Normality is not transitive
Normalizer-closedness
This subgroup property is normalizer-closed: the normalizer of any subgroup with this property, in the whole group, again has this property
View a complete list of normalizer-closed subgroup properties
The normalizer of a pronormal subgroup of a group is pronormal. In fact, it is an abnormal subgroup -- a stronger condition. For full proof, refer: Normalizer of pronormal implies abnormal, Pronormality is normalizer-closed
Normalizing joins
This subgroup property is normalizing join-closed: the join of two subgroups with the property, one of which normalizes the other, also has the property.
View other normalizing join-closed subgroup properties
If are pronormal subgroups of a group such that , then the join is also a pronormal subgroup. For full proof, refer: Pronormality is normalizing join-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 pronormal subgroups need not be pronormal. In fact, a join of finitely many pronormal subgroups need not be pronormal. For full proof, refer: Pronormality is not finite-join-closed
Upper join-closedness
NO: This subgroup property is not upper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.
If is a subgroup of and are intermediate subgroups containing such that is pronormal in both and , it is not necessary that is pronormal in .
If, for a subgroup of a group , there exists a unique largest subgroup in which is pronormal, is termed a pronormalizer for , and is termed a subgroup having a pronormalizer.
For full proof, refer: Pronormality is not upper join-closed
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
Suppose is a surjective homomorphism of groups. Then, if is a pronormal subgroup of , is a pronormal subgroup of . For full proof, refer: Pronormality satisfies image condition
Quotient-transitivity
This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties
If such that is a normal (hence also pronormal) subgroup of and is pronormal in , then is a pronormal subgroup of . For full proof, refer: Pronormality is quotient-transitive
Effect of property operators
The subordination
Applying the subordination to this property gives: subpronormal subgroup
A subgroup of a group is termed a subpronormal subgroup of if there exists a sequence with each a pronormal subgroup in . Any subgroup of a finite group is subpronormal.
The right transiter
Applying the right transiter to this property gives: right-transitively pronormal subgroup
A subgroup of a group is termed right-transitively pronormal in if any pronormal subgroup of is pronormal in . Any SCAB-subgroup is right-transitively pronormal.
The join-transiter
Applying the join-transiter to this property gives: join-transitively pronormal subgroup
A subgroup of a group is termed join-transitively pronormal in if the join of with any pronormal subgroup of is pronormal.
The hereditarily operator
Applying the hereditarily operator to this property gives: hereditarily pronormal subgroup
A subgroup of a group is termed hereditarily pronormal if every subgroup of is pronormal in . Note that this is equivalent to being a right-transitively pronormal subgroup that is also a group in which every subgroup is pronormal.
The join-closure
Applying the join-closure to this property gives: join of pronormal subgroups
A subgroup of a group is termed a join of pronormal subgroups in if there is a set of pronormal subgroups of whose join in .
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: IsPronormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
While there is no built-in GAP command for testing pronormality, the test can be accomplished by a short piece of GAP code, available at GAP:IsPronormal. The code is invoked as follows:
IsPronormal(group,subgroup);
References
Textbook references
Book | Page number | Chapter and section | Contextual information | View |
---|---|---|---|---|
Finite Groups by Daniel Gorenstein, ISBN 0821843427^{More info} | 13 | Chapter 1, Exercise 4 | definition introduced in exercise | Google Books |
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info} | 298 | Section 10.4 | Google Books (page preview unavailable) |
Journal references
- Finite soluble groups with pronormal system normalizers by John S. Rose, Proceedings of the London Mathematical Society, ISSN 1460244X (online), ISSN 00246115 (print), Volume 17, Page 447 - 469(Year 1967): ^{Weblink on Oxford Journals page}^{More info}
- On the system normalizers of a soluble group by Philip Hall, Page 507 - 528(Year 1938): ^{}^{More info}
- Pronormality in finite groups by T. A. Peng, Journal of the London Mathematical Society, ISSN 14697750 (online), ISSN 00246107 (print), Volume 3, Page 301 - 306(Year 1971): ^{Weblink}^{More info}
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