# Transfer-closed fully invariant subgroup

From Groupprops

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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]

## Definition

A subgroup of a group is termed **transfer-closed fully invariant** if, for any subgroup of , is a fully invariant subgroup of .

## Formalisms

### In terms of the transfer condition operator

This property is obtained by applying the transfer condition operator to the property: fully invariant subgroup

View other properties obtained by applying the transfer condition operator

## Relation with other properties

### Stronger properties

### Weaker properties

- Intermediately fully invariant subgroup
- Fully invariant subgroup
- Transfer-closed characteristic subgroup
- Intermediately characteristic subgroup
- Characteristic subgroup
- Normal subgroup

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT 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 transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### 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

### Intersection-closedness

YES:This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.In fact, since the property is also true for every group as a subgroup of itself, it is a strongly intersection-closed subgroup property.ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closedABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness