Direct product-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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This article is about a general term. A list of important particular cases (instances) is available at Category:Direct product-closed subgroup properties

Definition

Definition with symbols

A subgroup property $p$ is termed direct product-closed if, given a nonempty indexing set $I$, and a collection of group-subgroup pairs $H_{i}\leq G_{i}$ for $i\in I$, such that each $H_{i}$ satisfies $p$ as a subgroup of $G_{i}$, then the direct product of the $H_{i}$s satisfies $p$ via its natural embedding in the direct product of $G_{i}$s.