Pronormal subgroup

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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 $H$ 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 $g H g - 1$ , there exists $x \in K$ such that $x H x - 1 = g H g - 1$ .
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 $H g = g - 1 H g$ and the angled braces are for the join of subgroups (or subgroup generated).	for any $g \in G$ , there exists $x \in \langle H, gHg^{-1} \rangle$ such that $g H g - 1 = x H x - 1$ .

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

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.

1 Definition
- 1.1 Symbol-free definition
- 1.2 Equivalence of definitions
2 History
- 2.1 Origin
3 Examples
4 Formalisms
- 4.1 Monadic second-order description
5 Relation with other properties
6 Metaproperties
7 Effect of property operators
8 Testing
- 8.1 GAP code
9 References
- 9.1 Textbook references
- 9.2 Journal references
10 External links
- 10.1 Definition links

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.[SHOW MORE]

VIEW: Definitions built on this | Facts about this | Survey articles about this | Facts about definitions built on this |
VIEW RELATED:
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.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions [SHOW MORE]

VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions |
VIEW FACTS THAT: use, or start with, a subgroup satisfying the property| prove, or show that, a subgroup satisfies the property
RANDOM SUBGROUP PROPERTY: 2-subnormal subgroup: A normal subgroup of a normal subgroup. Need not be normal, because normality is not transitive.

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 |
VIEW: |

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 $H$ 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

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	click here	normal versus pronormal
join-transitively pronormal subgroup	join with any pronormal subgroup is pronormal		pronormality is not finite-join-closed		--
right-transitively pronormal subgroup	any pronormal subgroup of it is pronormal in the whole group		pronormality is not transitive		--
abnormal subgroup	$g \in \langle H,H^g \rangle$ for all $g$	abnormal implies pronormal	pronormal not implies abnormal	click here	--
maximal subgroup	proper subgroup and no other proper subgroup containing it	maximal implies pronormal		click here	--
procharacteristic subgroup		procharacteristic implies pronormal	pronormal not implies procharacteristic		subnormal-to-normal and normal-to-characteristic
* intermediately isomorph-conjugate subgroup	isomorph-conjugate subgroup in every intermediate subgroup			click here	subnormal-to-normal and normal-to-characteristic
Sylow subgroup		Sylow implies pronormal		click here	--
procharacteristic subgroup of normal subgroup		procharacteristic of normal implies pronormal			--
* intermediately isomorph-conjugate subgroup of normal subgroup		intermediately isomorph-conjugate of normal implies pronormal			--
* Sylow subgroup of normal subgroup		Sylow of normal implies pronormal			--
* Hall subgroup of solvable normal subgroup					--
* nilpotent Hall subgroup of normal subgroup					--
subgroup whose join with any distinct conjugate is the whole group		join with any distinct conjugate is the whole group implies pronormal			--
* 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 |

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		click here
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		click here
weakly pronormal subgroup		pronormal implies weakly pronormal	weakly pronormal not implies pronormal
paranormal subgroup		pronormal implies paranormal	paranormal not implies pronormal
self-conjugate-permutable subgroup		pronormal implies self-conjugate-permutable	self-conjugate-permutable not implies pronormal
join of pronormal subgroups	join of (possibly infinitely many) pronormal subgroups	(by definition)	pronormality is not finite-join-closed
join of finitely many pronormal subgroups	join of finitely many pronormal subgroups	(by definition)	pronormality is not finite-join-closed
subgroup with abnormal normalizer	the normalizer is an abnormal subgroup	normalizer of pronormal implies abnormal		click here
subgroup with weakly abnormal normalizer	the normalizer is a self-normalizing subgroup			click here
subgroup with self-normalizing normalizer	the normalizer is a self-normalizing subgroup			click here
WNSCDIN-subgroup		pronormal implies WNSCDIN\|WNSCDIN not implies pronormal	click here
MWNSCDIN-subgroup		Pronormal implies MWNSCDIN	MWNSCDIN not implies pronormal
subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed		via WNSCDIN	(via WNSCDIN)	click here
subgroup having a pronormalizer	has a pronormalizer
subgroup having a subnormalizer	has a subnormalizer			click here
subpronormal subgroup	obtained by applying the subordination operator to pronormality

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: |
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 $H \le G$ is pronormal and $K \le G$ , 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: |
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 $K \le N_G(H)$ , then the join $H K$ 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 $H \le K \le G$ 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 $H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G$ 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

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}

External links

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Definition links

Facts about Pronormal subgroupRDF feed

Applying operator gives	Subpronormal subgroup +, Right-transitively pronormal subgroup +, Join-transitively pronormal subgroup +, Hereditarily pronormal subgroup +, and Join of pronormal subgroups +
Defined in	Book:Gorenstein (13, Chapter 1, Exercise 4, definition introduced in exercise) +, and Book:RobinsonGT (298, Section 10.4, ?) +
Defining ingredient	Conjugate subgroups +, and Join of subgroups +
Dissatisfies metaproperty	Intersection-closed subgroup property +, Strongly intersection-closed subgroup property +, Transitive subgroup property +, and Join-closed subgroup property +
Page class	Term +
Particular example	Symmetric group:S3 +, and Symmetric group:S4 +
Referenced in	Book:Gorenstein (13, Chapter 1, Exercise 4, definition introduced in exercise) +, Book:RobinsonGT (298, Section 10.4, ?) +, Paper:Rose-pronormal (?, ?, ?) +, Paper:Hallonsystemnormalizers (?, ?, ?) +, Paper:Peng-pronormal (?, ?, ?) +, Springer Online Reference Works (?, ?, ?) +, planetmath (?, ?, ?) +, and Wikipedia (?, ?, ?) +
Satisfies metaproperty	Monadic second-order subgroup property +, Trim subgroup property +, Trivially true subgroup property +, Identity-true subgroup property +, Left-realized subgroup property +, Right-realized subgroup property +, Intermediate subgroup condition +, Normalizing join-closed subgroup property +, Image condition +, and Quotient-transitive subgroup property +
Stronger than	Subnormal-to-normal subgroup +, Intermediately subnormal-to-normal subgroup +, Weakly pronormal subgroup +, Paranormal subgroup +, Self-conjugate-permutable subgroup +, Join of pronormal subgroups +, Join of finitely many pronormal subgroups +, Subgroup with abnormal normalizer +, Subgroup with weakly abnormal normalizer +, Subgroup with self-normalizing normalizer +, WNSCDIN-subgroup +, MWNSCDIN-subgroup +, Subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed +, Subgroup having a pronormalizer +, Subgroup having a subnormalizer +, and Subpronormal subgroup +
Term introduced by	Hall +
Variation of	Normal subgroup +
Weaker than	Normal subgroup +, Join-transitively pronormal subgroup +, Right-transitively pronormal subgroup +, Abnormal subgroup +, Maximal subgroup +, Procharacteristic subgroup +, Intermediately isomorph-conjugate subgroup +, Sylow subgroup +, Procharacteristic subgroup of normal subgroup +, Intermediately isomorph-conjugate subgroup of normal subgroup +, Sylow subgroup of normal subgroup +, Hall subgroup of solvable normal subgroup +, Nilpotent Hall subgroup of normal subgroup +, Subgroup whose join with any distinct conjugate is the whole group +, and Subgroup whose join with any distinct conjugate is a SCAB-subgroup +