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 $H_i \le G_i$ 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.