# Transfer condition

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

This article is about a general term. A list of important particular cases (instances) is available at Category:Subgroup properties satisfying transfer condition

## Definition

### Definition with symbols

A subgroup property is said to satisfy the **transfer condition** if whenever satisfies property as a subgroup of , and is a subgroup of , then ∩ satisfies property as a subgroup of .

## Formalisms

This article defines a single-input-expressible subgroup metaproperty

Consider the procedure that takes as input a group-subgroup pair , and outputs all group-subgroup pairs for . The transfer condition is the single-input-expressible metaproperty corresponding to procedure : a subgroup property satisfies the transfer condition if satisfying property implies that all pairs also satisfy property .

## Relation with other metaproperties

### Stronger metaproperties

### Weaker metaproperties

### Conjunction implications

- Any transitive subgroup property that satisfies the transfer condition is also finite-intersection-closed.
`For full proof, refer: Transitive and transfer condition implies finite-intersection-closed`

## Metametaproperties

### Conjunction-closedness

This subgroup metaproperty is conjunction-closed: an arbitrary conjunction (AND) of subgroup properties satisfying this metaproperty, also satisfies this metaproperty

View conjunction-closed subgroup metaproperties

### Disjunction-closedness

This subgroup metaproperty is disjunction-closed: an arbitrary disjunction (OR) of subgroup properties satisfying this metaproperty, also satisfies this metaproperty

View all disjunction-closed subgroup metaproperties

### Composition-closedness

This subgroup metaproperty is composition-closed: the property obtained by applying the composition operator to two subgroup properties satisfying this metaproperty, also satisfies this metaproperty

View a complete list of composition-closed subgroup metaproperties