# 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 $H$ 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 (or 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 $H^g = gHg^{-1}$. 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.

### 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.

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 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

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

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 quick description 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) Join-transitively pronormal subgroup, Subgroup whose join with any distinct conjugate is the whole group, Transfer-closed pronormal subgroup|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 Join-transitively pronormal subgroup|FULL LIST, MORE INFO --
Maximal subgroup Proper subgroup and no other proper subgroup containing it maximal implies pronormal (see also list of examples) Subgroup whose join with any distinct conjugate is the whole group|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 Procharacteristic subgroup|FULL LIST, MORE INFO subnormal-to-normal and normal-to-characteristic
Sylow subgroup Sylow implies pronormal Intermediately isomorph-conjugate subgroup, Nilpotent Hall subgroup, Nilpotent pronormal subgroup, Sylow subgroup of normal subgroup|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

## 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 $H \le G$ 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: 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.

### 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 $HK$ 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 $H \le K \le G$ 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
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)