# Transfer condition

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
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## Definition

### Definition with symbols

A subgroup property $p$ is said to satisfy the transfer condition if whenever $H$ satisfies property $p$ as a subgroup of $G$, and $K$ is a subgroup of $G$, then $H$$K$ satisfies property $p$ as a subgroup of $K$.

## Formalisms

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

## Metametaproperties

### Conjunction-closedness

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

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