Pronormal 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.
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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]

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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

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.

Definition

Symbol-free definition

A subgroup of a group is termed pronormal if it satisfies the following equivalent conditions:

• Any conjugate subgroup of the subgroup inside the whole group is also conjugate inside any intermediate subgroup
• Any conjugate subgroup of the subgroup is conjugate to it inside the subgroup generated by the original subgroup and its conjugate.

Definition with symbols

A subgroup $H$ of a group $G$ is termed pronormal if it satisfies the following equivalent conditions:

• For any $g\in G$ and any $K$ containing both $H$ and $H^{g}$, there exists $x\in K$ such that $H^{g}=H^{x}$.
• (right-action convention): For any $g$ in $G$, there exists $x\in \langle H,H^{g}\rangle$ such that $H^{g}=H^{x}$. Here $H^{g}=g^{-1}Hg$ denotes the conjugate subgroup of $H$ by the element $g$ (acting on the right, or $g^{-1}$ acting on the left), and the angled braces are for the join of subgroups (or subgroup generated).
• (left-action convention): For any $g\in G$, there exists $x\in \langle H,gHg^{-1}\rangle$ such that $gHg^{-1}=xHx^{-1}$.

Equivalence of definitions

The two definitions are equivalent because being conjugate inside the smallest possible intermediate subgroup, viz $\langle H,H^{g}\rangle$, implies being conjugate in any intermediate subgroup.

Examples

If you're interested in pronormal subgroups in a particular group, view the article on that particular group and hunt for the subsection titled Pronormal subgroups

Extreme examples

1. Every group is pronormal as a subgroup of itself
2. The trivial subgroup is always pronormal.

Generic examples

1. All Sylow subgroups are pronormal.
2. Maximal subgroups and normal subgroups are pronormal.

Particular examples

1. High occurence example: In the symmetric group of order three, all subgroups are pronormal.
2. 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

1. 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.
2. 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 $H$ is a subgroup of $G$, the group $\langle H,H^{g}\rangle$ can be constructed using monadic second-order logic. Pronormality testing is now the following sentence:

$\forall g\in G\ \exists x\in \langle H,H^{g}\rangle :\ (\forall y\in H\exists z\in H.gyg^{-1}=zxz^{-1})$

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

• Normal subgroup: For proof of the implication, refer normal implies pronormal and for proof of its strictness (i.e. the reverse implication being false) refer pronormal not implies normal.View a survey article comparing and contrasting these terms, at: normal versus pronormal.For intermediate notions between pronormal subgroup and normal subgroup, click here.
• Abnormal subgroup
• Maximal subgroup
• Intermediately isomorph-conjugate subgroup: For full proof, refer: Intermediately isomorph-conjugate implies pronormal
• Intermediately isomorph-conjugate subgroup of normal subgroup: For full proof, refer: Intermediately isomorph-conjugate of normal implies pronormal
• Sylow subgroup in a finite group
• Sylow subgroup of normal subgroup: For full proof, refer: Sylow of normal implies pronormal
• Join-transitively pronormal subgroup
• Subgroup whose join with any distinct conjugate is the whole group: For full proof, refer: Join with any distinct conjugate is the whole group implies pronormal
• Procharacteristic subgroup

View a more comprehensive list of subgroup properties stronger than pronormality

Weaker properties

• 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
• 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
• Subgroup with abnormal normalizer: For full proof, refer: Normalizer of pronormal implies abnormal
• Subgroup with weakly abnormal normalizer
• 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.
• Subnormal-to-normal subgroup: For full proof, refer: Pronormal and subnormal implies normal
• Subgroup with self-normalizing normalizer
• 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.
• 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.

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 $H\leq G$ is pronormal and $K\leq G$, then $H\cap K$ need not be pronormal in $K$. 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. For full proof, refer: Pronormality is not 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. The subordination of this property is the property of being subpronormal. Further information: Pronormality 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 $H,K$ are pronormal subgroups of a group $G$ such that $K\leq N_{G}(H)$, then the join $HK$ 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 $H$ is a subgroup of $G$ and $K,L$ are intermediate subgroups containing $H$ such that $H$ is pronormal in both $K$ and $L$, it is not necessary that $H$ is pronormal in $\langle K,L\rangle$. 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 $\varphi :G\to K$ is a surjective homomorphism of groups. Then, if $H$ is a pronormal subgroup of $G$, $\varphi (H)$ is a pronormal subgroup of $K$. 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 $H\leq K\leq G$ such that $H$ is a normal subgroup of $G$ and $K/H$ is pronormal in $G/H$, then $K$ is a pronormal subgroup of $G$. For full proof, refer: Pronormality is quotient-transitive

Facts

The normalizer of any pronormal subgroup is abnormal. For full proof, refer: Normalizer of pronormal implies abnormal

Effect of property operators

The subordination

Applying the subordination to this property gives: subpronormal subgroup

A subgroup $H$ of a group $G$ is termed a subpronormal subgroup of $G$ if there exists a sequence $H=H_{0}\leq H_{1}\leq H_{2}\leq \dots \leq H_{n}=G$ with each $H_{i-1}$ a pronormal subgroup in $H_{i}$.

The right transiter

Applying the right transiter to this property gives: right-transitively pronormal subgroup

A subgroup $H$ of a group $G$ is termed right-transitively pronormal in $G$ if any pronormal subgroup of $H$ is pronormal in $G$. Any SCAB-subgroup is right-transitively pronormal.

The join-transiter

Applying the join-transiter to this property gives: join-transitively pronormal subgroup

A subgroup $H$ of a group $G$ is termed join-transitively pronormal in $G$ if the join of $H$ with any pronormal subgroup of $G$ is pronormal.

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: IsPronormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

GAP-codable subgroup property

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

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 13, Chapter 1, Exercise 4 (definition introduced in exercise)

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 pageMore info}
• On the system normalizers of a soluble group by Philip Hall, Page 507 - 528(Year 1938): ^{More info}

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