Pronormal subgroup

Definition

QUICK PHRASES: conjugates in whole group are conjugate in intermediate subgroups, conjugates in whole group are conjugate in join

Equivalent definitions in tabular format

No.	Shorthand	A subgroup of a group is pronormal if ...	A subgroup $H$ of a group $G$ is pronormal if... (right-action convention)	A subgroup $H$ of a group $G$ 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 $g\in G$ and any subgroup $K$ of $G$ containing both $H$ and $H^{g}$, there exists $x\in K$ such that $H^{g}=H^{x}$.	for any $g\in G$ and any subgroup $K$ of $G$ containing both $H$ and $gHg^{-1}$, there exists $x\in K$ such that $xHx^{-1}=gHg^{-1}$.
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 $g$ in $G$, there exists $x\in \langle H,H^{g}\rangle$ such that $H^{g}=H^{x}$. Here $H^{g}=g^{-1}Hg$ and the angled braces are for the join of subgroups (i.e., the subgroup generated).	for any $g\in G$, there exists $x\in \langle H,gHg^{-1}\rangle$ such that $gHg^{-1}=xHx^{-1}$.

Under the right action convention, the right action of $g$ on $H$ by conjugation gives the subgroup $H^{g}=g^{-1}Hg$. Under the left action convention, the left action of $g$ on $H$ by conjugation gives the subgroup ${}^{g}H=gHg^{-1}$. 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.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

Equivalence of definitions

The two definitions are equivalent because being conjugate inside the smallest possible intermediate subgroup, i.e., $\langle H,H^{g}\rangle$, 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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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

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 degree 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 $\langle H,gHg^{-1}\rangle$ can be constructed using monadic second-order logic. Pronormality testing is now the following sentence:

$\forall g\in G\ \exists x\in \langle H,gHg^{-1}\rangle :\ (\forall y\in H\exists z\in H.gyg^{-1}=xzx^{-1})\land (\forall v\in H\exists w\in H.gwg^{-1}=xvx^{-1})$

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions	Comparison
normal subgroup	equal to all conjugate subgroups	normal implies pronormal	pronormal not implies normal (see also list of examples)	|FULL LIST, MORE INFO	normal versus pronormal
join-transitively pronormal subgroup	join with any pronormal subgroup is pronormal		pronormality is not finite-join-closed	|FULL LIST, MORE INFO	--
right-transitively pronormal subgroup	any pronormal subgroup of it is pronormal in the whole group		pronormality is not transitive	|FULL LIST, MORE INFO	--
abnormal subgroup	$g\in \langle H,H^{g}\rangle$ for all $g$	abnormal implies pronormal	pronormal not implies abnormal (see also list of examples)	|FULL LIST, MORE INFO	--
maximal subgroup	proper subgroup and no other proper subgroup containing it	maximal implies pronormal	(see also list of examples)	|FULL LIST, MORE INFO	--
procharacteristic subgroup		procharacteristic implies pronormal	pronormal not implies procharacteristic	|FULL LIST, MORE INFO	subnormal-to-normal and normal-to-characteristic
intermediately isomorph-conjugate subgroup	isomorph-conjugate subgroup in every intermediate subgroup			|FULL LIST, MORE INFO	subnormal-to-normal and normal-to-characteristic
Sylow subgroup		Sylow implies pronormal		|FULL LIST, MORE INFO	--
procharacteristic subgroup of normal subgroup		procharacteristic of normal implies pronormal		|FULL LIST, MORE INFO	--
* intermediately isomorph-conjugate subgroup of normal subgroup		intermediately isomorph-conjugate of normal implies pronormal		|FULL LIST, MORE INFO	--
* Sylow subgroup of normal subgroup		Sylow of normal implies pronormal		|FULL LIST, MORE INFO	--
* Hall subgroup of solvable normal subgroup				|FULL LIST, MORE INFO	--
* nilpotent Hall subgroup of normal subgroup				|FULL LIST, MORE INFO	--
subgroup whose join with any distinct conjugate is the whole group		join with any distinct conjugate is the whole group implies pronormal		|FULL LIST, MORE INFO	--
* subgroup whose join with any distinct conjugate is a SCAB-subgroup

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

For a survey article exploring these properties in greater depth, refer: subnormal-to-normal and normal-to-characteristic

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subnormal-to-normal subgroup	if the subgroup is subnormal in the whole group, it is normal in the whole group	pronormal and subnormal implies normal		|FULL LIST, MORE INFO
intermediately subnormal-to-normal subgroup	if the subgroup is subnormal in an intermediate subgroup, it is also normal in that intermediate subgroup	pronormal implies intermediately subnormal-to-normal		|FULL LIST, MORE INFO
weakly pronormal subgroup		pronormal implies weakly pronormal	weakly pronormal not implies pronormal	|FULL LIST, MORE INFO
paranormal subgroup		pronormal implies paranormal	paranormal not implies pronormal	|FULL LIST, MORE INFO
self-conjugate-permutable subgroup		pronormal implies self-conjugate-permutable	self-conjugate-permutable not implies pronormal	|FULL LIST, MORE INFO
join of pronormal subgroups	join of (possibly infinitely many) pronormal subgroups	(by definition)	pronormality is not finite-join-closed	|FULL LIST, MORE INFO
join of finitely many pronormal subgroups	join of finitely many pronormal subgroups	(by definition)	pronormality is not finite-join-closed	|FULL LIST, MORE INFO
subgroup with abnormal normalizer	the normalizer is an abnormal subgroup	normalizer of pronormal implies abnormal		|FULL LIST, MORE INFO
subgroup with weakly abnormal normalizer	the normalizer is a self-normalizing subgroup			|FULL LIST, MORE INFO
subgroup with self-normalizing normalizer	the normalizer is a self-normalizing subgroup			|FULL LIST, MORE INFO
WNSCDIN-subgroup		pronormal implies WNSCDIN	WNSCDIN not implies pronormal	|FULL LIST, MORE INFO
MWNSCDIN-subgroup		Pronormal implies MWNSCDIN	MWNSCDIN not implies pronormal	|FULL LIST, MORE INFO
subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed		via WNSCDIN	(via WNSCDIN)	|FULL LIST, MORE INFO
subgroup having a pronormalizer	has a pronormalizer			|FULL LIST, MORE INFO
subgroup having a subnormalizer	has a subnormalizer			|FULL LIST, MORE INFO
subpronormal subgroup	obtained by applying the subordination operator to pronormality			|FULL LIST, MORE INFO

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. 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 $H,K$ 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 $\langle K,L\rangle$.

If, for a subgroup $H$ of a group $G$, there exists a unique largest subgroup $K$ in which $H$ is pronormal, $K$ is termed a pronormalizer for $H$, and $H$ 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 $\varphi :G\to K$ 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 (hence also pronormal) 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

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 $H_{i}$. 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 $H$ 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.

The hereditarily operator

Applying the hereditarily operator to this property gives: hereditarily pronormal subgroup

A subgroup $H$ of a group $G$ is termed hereditarily pronormal if every subgroup of $H$ is pronormal in $G$. 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 $H$ of a group $G$ is termed a join of pronormal subgroups in $G$ if there is a set of pronormal subgroups of $G$ whose join in $H$.

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

Book	Page number	Chapter and section	Contextual information	View
Finite Groups by Daniel Gorenstein, ISBN 0821843427More 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 0387944613More 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 pageMore 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): ^{WeblinkMore info}