Pronormal subgroup

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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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

Symbol-free definition

A subgroup of a group is termed pronormal if any conjugate of the subgroup inside the whole group is also conjugate inside any intermediate subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed pronormal if for any $g$ in $G$ , there exists $x\in <H,H^{g}>$ such that $H^{g}=H^{x}$ .

Relation with other properties

Stronger properties

Weaker properties=

Metaproperties

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.