# 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 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

Metaproperty name | Satisfied? | Proof | Difficulty level | Statement with symbols |
---|---|---|---|---|

transitive subgroup property | Yes | transfer-closed characteristicity is transitive | If are groups such that is transfer-closed characteristic in and is transfer-closed characteristic in , then is transfer-closed characteristic in . | |

intermediate subgroup condition | Yes | If are groups such that is transfer-closed characteristic in , then is transfer-closed characteristic in . | ||

strongly intersection-closed subgroup property | Yes | If is a family of transfer-closed characteristic subgroups of a group , the intersection is also a transfer-closed characteristic subgroup of . | ||

transfer condition | Yes | If and are subgroups of such that is transfer-closed characteristic in , then is transfer-closed characteristic in . |