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

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.

Relation with other properties

Stronger properties

• Normal subgroup
• Abnormal subgroup
• Maximal subgroup
• Intermediately isomorph-conjugate subgroup
• Sylow subgroup in a finite group

Weaker properties

• Weakly pronormal subgroup
• Paranormal subgroup
• Polynormal subgroup
• Subgroup with abnormal normalizer

Conjunction with other properties

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

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.

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