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 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 $H^{g}=H^{x}$ .
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).

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

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 order 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 $\langle H,H^{g}\rangle$ 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

Stronger properties

Normal subgroup
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

Weaker properties

Conjunction with other properties

Any pronormal subnormal subgroup is normal: For full proof, refer: Pronormal and subnormal implies normal

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)

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

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

Facts

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

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

GAP-codable subgroup property

There isn't any in-built GAP command for pronormality, but the following simple GAP code can be used to test pronormality:

IsPronormal := function(G,H)
	    local K,g,x,flag;
	    for g in Set(G) do
	    	K := Group(Union(H,H^g));
		flag := false;
		for x in K do
		    if H^g = H^x then flag:=true; fi;
		 od;
		 if (flag = false) then return false; fi;
		od;
	return true;
end;;

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 page}^{More info}

External links

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