Transfer condition

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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
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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 $p$ 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

This article defines a single-input-expressible subgroup metaproperty

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

Relation with other metaproperties

Stronger metaproperties

• Inverse image condition
• Strongly UL-intersection-closed subgroup property

Weaker metaproperties

• Intermediate subgroup condition

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