Polynormal subgroup
From Groupprops
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
Contents
Definition
Definition with symbols
A subgroup of a group is termed polynormal if given any , is a contranormal subgroup in the subgroup , i.e., the closure of under the action by conjugation of the cyclic subgroup generated by .
Relation with other properties
Stronger properties
- Normal subgroup
- Maximal subgroup
- Abnormal subgroup
- Pronormal subgroup
- Weakly abnormal subgroup
- Weakly pronormal subgroup
- Strongly paranormal subgroup
- Paranormal subgroup
- Strongly polynormal subgroup
- Sylow subgroup in a finite group
Weaker properties
- Fan subgroup
- Intermediately subnormal-to-normal subgroup: For full proof, refer: Polynormal implies intermediately subnormal-to-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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ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
If is polynormal in , is also polynormal in any intermediate subgroup . For full proof, refer: Polynormality satisfies 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).
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The whole group and the trivial subgroup are polynormal; in fact they are normal.
Join-closedness
YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
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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 polynormal subgroups is polynormal. For full proof, refer: Polynormality 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: IsPolynormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
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