# Intermediately fully invariant 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]

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed **intermediately fully invariant** or **intermediately fully characteristic** if it is fully invariant in every intermediate subgroup of the group containing it.

### Definition with symbols

A subgroup of a group is termed **intermediately fully invariant** or **intermediately fully characteristic** in if, for any intermediate subgroup of , is fully characteristic in : for any endomorphism of , .

## Relation with other properties

### Stronger properties

- Homomorph-containing subgroup
- Subhomomorph-containing subgroup
- Variety-containing subgroup
- Normal Sylow subgroup
- Normal Hall subgroup
- Normal subgroup having no nontrivial homomorphism to its quotient group

### Weaker 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

An intermediately fully characteristic subgroup of an intermediately fully characteristic subgroup need not be intermediately fully characteristic.

`For full proof, refer: Intermediate full invariance is not transitive`

### Join-closedness

YES:This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.In fact, since the property is also true for the trivial subgroup in any group, it is a strongly join-closed subgroup property.ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closedABOUT 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

An arbitrary join of intermediately fully characteristic subgroups is intermediately fully characteristic. This follows from the fact that the intermediately operator preserves the property of being closed under joins.

`For full proof, refer: Intermediate full invariance is strongly join-closed`

`Further information: Intermediately operator preserves join-closedness, Full invariance is strongly join-closed`

### 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

### 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 full invariance is quotient-transitive`

`Further information: Intermediately operator preserves quotient-transitivity, Full invariance is quotient-transitive`