Finite-relative-intersection-closed subgroup property

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

A subgroup property $p$ is termed finite-relative-intersection-closed if it satisfies the following:

Suppose $H,K$ are subgroups of a group $G$ such that $H$ satisfies property $p$ in $G$ and $K$ satisfies property $p$ in some subgroup of $G$ containing both $H$ and $K$. Then, $H\cap K$ satisfies property $p$ in $G$.

Relation with other metaproperties

Stronger metaproperties

• Relative-intersection-closed subgroup property

Weaker metaproperties

• Finite-intersection-closed subgroup property
• Transitive subgroup property

Facts

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