UL-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 UL-intersection-closed if for any group $G$, any nonempty (but otherwise arbitrary, possibly infinite) indexing set $I$, and subgroups $H_{i}\leq K_{i}\leq G,i\in I$, such that $H_{i}$ satisfies property $p$ in $K_{i}$ for each $i\in I$, we have:

$\bigcap _{i\in I}H_{i}$ satisfies property $p$ in $\bigcap _{i\in I}K_{i}$.