# Transfer-closed characteristic subgroup

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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 $H$ of a group $G$ is termed a transfer-closed characteristic subgroup if, for any subgroup $K \le G$, $H \cap K$ is a characteristic subgroup of $K$.

## 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

## 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 transitive
ABOUT 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 $H \le K \le G$ are groups such that $K$ is a transfer-closed characteristic subgroup of $G$ and $H$ is a transfer-closed characteristic subgroup of $K$. Then, $H$ is a transfer-closed characteristic subgroup of $G$.

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 condition
ABOUT 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