Paranormal subgroup
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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]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality
Definition
Definition with symbols
(right-action convention): A subgroup of a group is termed paranormal if for any , is a contranormal subgroup of ; in other words, the normal closure of in is .
Here is a conjugate of , and the angled braces are for the subgroup generated.
(left-action convention): A subgroup of a group is termed paranormal if for any , is a contranormal subgroup of ; in other words, the normal closure of in is .
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.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
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
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.GAP-codable subgroup property
View the GAP code for testing this subgroup property at: IsParanormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
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