Paranormal 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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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed paranormal if for any $g\in G$, $H$ is a contranormal subgroup of $⟨H,H^g⟩$; in other words, the normal closure of $H$ in $⟨H,H^g⟩$ is $⟨H,H^g⟩$.

Here $H^g=g^{-1}Hg$ is a conjugate of $H$, and the angled braces are for the subgroup generated.

Relation with other properties

Stronger properties

• Normal subgroup
• Pronormal subgroup
• Abnormal subgroup
• Join of pronormal subgroups
• Strongly paranormal subgroup: For proof of the implication, refer Strongly paranormal implies paranormal and for proof of its strictness (i.e. the reverse implication being false) refer Paranormal not implies strongly paranormal.
• Weakly abnormal subgroup

Weaker properties

• Polynormal subgroup: It has been conjectured that for finite groups, the two notions coincide; however this has neither been proved nor disproved.

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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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

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

In fact, an arbitrary, possibly empty, join of paranormal subgroups is paranormal. For full proof, refer: Paranormality is strongly join-closed

References

• On the arrangement of intermediate subgroups by M. S. Ba and Z. I. Borevich
• On the arrangement of subgroups by Z. I. Borevich, Zap. Nauchn. Semin. tOMI, 94, 5-12 (1979)
• On the lattice of subgroups by Z. I. Borevich and O. N. Macedonska, Zap. Nauchn. Semin. LOMI, 103, 13-19, 1980
• Testing of subgroups of a finite group for some embedding properties like pronormality by V. I. Mysovskikh