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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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 $H^{g}=gHg^{-1}$ denotes the conjugate subgroup of $H$ by the element $g$ , 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 $<H,H^{g}>$ , implies being conjugate in any intermediate subgroup.

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
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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 $<H,H^{g}>$ can be constructed using monadic second-order logic. Pronormality testing is now the following sentence:

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

Relation with other properties

Stronger properties

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.
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Pronormality satisfies the intermediate subgroup condition, that is, any pronormal subgroup is pronormal in every intermediate subgroup.

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. This statement needs to be verified.

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
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The property of pronormality is not transitive. The subordination of this property is the property of being subpronormal. This statement needs to be verified.

Facts

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

References

Finite soluble groups with pronormal system normalizers by John S. Rose, Proceedings of the London Mathematical Society(3) 17 (1967), 447-69

External links

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