Intermediately characteristic subgroup: Difference between revisions
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* [[Weaker than::Isomorph-free subgroup]] | * [[Weaker than::Isomorph-free subgroup]] | ||
* [[Weaker than::Order-unique subgroup]] | * [[Weaker than::Order-unique subgroup]] | ||
* [[Weaker than::Transfer-closed characteristic subgroup]]: {{proofofstrictimplicationat|[[Transfer-closed characteristic implies intermediately characteristic]]|[[intermediately characteristic not implies transfer-closed characteristic]]}} | |||
===Weaker properties=== | ===Weaker properties=== | ||
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{{proofat|[[Intermediate characteristicity is not transitive]]}} | {{proofat|[[Intermediate characteristicity is not transitive]]}} | ||
{{trim}} | {{trim}} | ||
{{quot-transitive}} | |||
== | {{proofat|[[Intermediate characteristicity is quotient-transitive]]}} | ||
==Effect of property operators== | |||
===Right transiter=== | ===Right transiter=== | ||
It turns out that any intermediately characteristic subgroup of a [[transfer-characteristic subgroup]] is again intermediately characteristic. This follows from some simple reasoning and the fact that [[characteristicity]] is itself [[transitive subgroup property|transitive]]. | It turns out that any intermediately characteristic subgroup of a [[transfer-closed characteristic subgroup]] is again intermediately characteristic. This follows from some simple reasoning and the fact that [[characteristicity]] is itself [[transitive subgroup property|transitive]]. {{further|[[Intermediately characteristic of transfer-closed characteristic implies intermediately characteristic]]}} | ||
Hence, the right transiter of the property of being intermediately characteristic is weaker than the property of being transfer-characteristic. | Hence, the right transiter of the property of being intermediately characteristic is weaker than the property of being transfer-closed characteristic. | ||
Revision as of 22:08, 17 September 2008
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 characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Definition
Symbol-free definition
A subgroup of a group is said to be intermediately characteristic if it is characteristic not only in the whole group but also in every intermediate subgroup.
Definition with symbols
A subgroup of a group is said to be intermediately characteristic if forany intermediate subgroup (such that ), is characteristic in .
In terms of the intermediately operator
The subgroup property of being intermediately characteristic can be obtained by applying the intermediately operator to the subgroup property of being characteristic.
Relation with other properties
Stronger properties
- Isomorph-free subgroup
- Order-unique subgroup
- Transfer-closed characteristic subgroup: For proof of the implication, refer Transfer-closed characteristic implies intermediately characteristic and for proof of its strictness (i.e. the reverse implication being false) refer intermediately characteristic not implies transfer-closed characteristic.
Weaker properties
Related properties
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
For full proof, refer: Intermediate characteristicity is not transitive
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
Quotient-transitivity
This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties
For full proof, refer: Intermediate characteristicity is quotient-transitive
Effect of property operators
Right transiter
It turns out that any intermediately characteristic subgroup of a transfer-closed characteristic subgroup is again intermediately characteristic. This follows from some simple reasoning and the fact that characteristicity is itself transitive. Further information: Intermediately characteristic of transfer-closed characteristic implies intermediately characteristic
Hence, the right transiter of the property of being intermediately characteristic is weaker than the property of being transfer-closed characteristic.