Polynormal 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]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality
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.
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