Linear-bound join-transitively subnormal subgroup: Difference between revisions
(New page: {{wikilocal}} {{subgroup property}} ==Definition== A subgroup <math>H</math> of a group <math>G</math> is termed '''linear-bound join-transitively subnormal''' if there exists a natu...) |
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{{variation of|join-transitively subnormal subgroup}} | |||
==Definition== | ==Definition== | ||
A [[subgroup]] <math>H</math> of a group <math>G</math> is termed '''linear-bound join-transitively subnormal''' if there exists a natural number <math>n</math> such that, given any <math>k</math>-subnormal subgroup <math>K</math> of <math>G</math>, the join <math>\langle H, K \rangle</math> is <math>nk</math>-subnormal. | A [[subgroup]] <math>H</math> of a group <math>G</math> is termed '''linear-bound join-transitively subnormal''' if there exists a natural number <math>n</math> such that, given any <math>k</math>-[[defining ingredient::subnormal subgroup]] <math>K</math> of <math>G</math>, the join <math>\langle H, K \rangle</math> is <math>nk</math>-subnormal. | ||
Here, a <math>k</math>-subnormal subgroup is a subgroup whose [[defining ingredient::subnormal depth]] is at most <math>k</math>. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Weaker than::2-subnormal subgroup]]: For a 2-subnormal subgroup, we can set <math>n = 2</math>. {{proofat|[[2-subnormal implies join-transitively subnormal]]}} | * [[Weaker than::2-subnormal subgroup]]: For a 2-subnormal subgroup, we can set <math>n = 2</math>. {{proofat|[[2-subnormal implies join-transitively subnormal]]}} | ||
* [[Weaker than::Subnormal subgroup of finite index]] | * [[Weaker than::Subnormal subgroup of finite index]] | ||
* [[Weaker than::Intermediately linear-bound join-transitively subnormal subgroup]] | |||
* [[Weaker than::Linear-bound intermediately join-transitively subnormal subgroup]] | |||
* [[Weaker than::Asymptotically fixed-depth join-transitively subnormal subgroup]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
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* [[Stronger than::Polynomial-bound join-transitively subnormal subgroup]] | * [[Stronger than::Polynomial-bound join-transitively subnormal subgroup]] | ||
* [[Stronger than::Join-transitively subnormal subgroup]] | * [[Stronger than::Join-transitively subnormal subgroup]] | ||
* [[Stronger than::Subnormal subgroup]] | |||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 22:00, 7 August 2009
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]
If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |
This is a variation of join-transitively subnormal subgroup|Find other variations of join-transitively subnormal subgroup |
Definition
A subgroup of a group is termed linear-bound join-transitively subnormal if there exists a natural number such that, given any -subnormal subgroup of , the join is -subnormal.
Here, a -subnormal subgroup is a subgroup whose subnormal depth is at most .
Relation with other properties
Stronger properties
- Normal subgroup: For a normal subgroup, we can set . For full proof, refer: Join of normal and subnormal implies subnormal of same depth
- 2-subnormal subgroup: For a 2-subnormal subgroup, we can set . For full proof, refer: 2-subnormal implies join-transitively subnormal
- Subnormal subgroup of finite index
- Intermediately linear-bound join-transitively subnormal subgroup
- Linear-bound intermediately join-transitively subnormal subgroup
- Asymptotically fixed-depth join-transitively subnormal subgroup
Weaker properties
- Polynomial-bound join-transitively subnormal subgroup
- Join-transitively subnormal subgroup
- Subnormal subgroup
Metaproperties
Transitivity
NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity
This follows from the fact that normal subgroups are linear-bound join-transitively subnormal, but not all subnormal subgroups are linear-bound join-transitively subnormal. That in turn follows from the fact that subnormality is not finite-join-closed. For full proof, refer: Join of two 3-subnormal subgroups need not be subnormal, subnormality is not finite-join-closed
If are both linear-bound join-transitively subnormal with corresponding natural numbers , their join is linear-bound join-transitively subnormal with natural number . (A smaller natural number might also work for the join).