# 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

### 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 .

## Formalisms

### 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

The subgroup property of being **intermediately characteristic** can be obtained by applying the intermediately operator to the subgroup property of being characteristic.

## Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this propertyVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## 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 isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT 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`
`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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

## 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.