Intermediately characteristic subgroup
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]
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity
Definition with symbols
In terms of the intermediately operator
This property is obtained by applying the intermediately operator to the property: characteristic subgroup
View other properties obtained by applying the intermediately operator
VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Relation with other 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.
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
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
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 Further information: Intermediately operator preserves quotient-transitivity, Characteristicity is quotient-transitive
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
Effect of property operators
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.