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

(right-action convention): A subgroup $H$ of a group $G$ is termed paranormal if for any $g\in G$, $H$ is a contranormal subgroup of $\langle H,H^{g}\rangle$; in other words, the normal closure of $H$ in $\langle H,H^{g}\rangle$ is $\langle H,H^{g}\rangle$.

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

(left-action convention): A subgroup $H$ of a group $G$ is termed paranormal if for any $g\in G$, $H$ is a contranormal subgroup of $\langle H,gHg^{-1}\rangle$; in other words, the normal closure of $H$ in $\langle H,gHg^{-1}\rangle$ is $\langle H,gHg^{-1}\rangle$.

Relation with other properties

Stronger properties

• Normal subgroup
• Pronormal subgroup: For proof of the implication, refer Pronormal implies paranormal and for proof of its strictness (i.e. the reverse implication being false) refer Paranormal not implies pronormal.
• 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: For proof of the implication, refer Weakly abnormal implies paranormal and for proof of its strictness (i.e. the reverse implication being false) refer Paranormal not implies weakly abnormal.
• Paracharacteristic subgroup
• Sylow subgroup
• Sylow subgroup of normal subgroup
• Hall subgroup: For full proof, refer: Hall implies paranormal
• Hall subgroup of normal subgroup: For full proof, refer: Hall of normal implies paranormal
• Intermediately isomorph-conjugate subgroup
• Procharacteristic 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.
• Weakly normal subgroup: For proof of the implication, refer Paranormal implies weakly normal and for proof of its strictness (i.e. the reverse implication being false) refer Weakly normal not implies paranormal.
• Intermediately subnormal-to-normal subgroup
• Subnormal-to-normal subgroup

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).
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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

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 the GAP code for testing this subgroup property at: IsParanormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

GAP-codable subgroup property

There is no built-in GAP command to test paranormality, but a short piece of GAP code can achieve this. The code is available at GAP:IsParanormal.

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