# Transfer-closed characteristic 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]

This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

## Definition

### Definition with symbols

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

## Formalisms

### In terms of the transfer condition operator

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

View other properties obtained by applying the transfer condition operator

## Relation with other properties

### Stronger properties

- Normal Sylow subgroup
- Normal Hall subgroup
- Variety-containing subgroup
- Subisomorph-containing subgroup

### Weaker properties

## 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

Suppose are groups such that is a transfer-closed characteristic subgroup of and is a transfer-closed characteristic subgroup of . Then, is a transfer-closed characteristic subgroup of .

`For full proof, refer: Transfer-closed characteristicity is transitive`

`Further information: Transfer condition operator preserves transitivity, Characteristicity is 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

### 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

### Transfer condition

YES:This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.

View other subgroup properties satisfying the transfer condition