Join-closed subgroup property

Symbol-free definition
A subgroup property $$p$$ is termed join-closed if the join of a nonempty (but otherwise arbitrary, possibly infinite) collection of subgroups, each with property $$p$$, also has property $$p$$.

Definition with symbols
A subgroup property $$p$$ is termed join-closed if given a group $$G$$, a nonempty indexing set $$I$$, and a collection of subgroups $$H_i$$ for $$i \in I$$, such that each $$H_i$$ satisfies $$p$$, the join, i.e. the subgroup generated by the $$H_i$$s, also satisfies $$p$$.

Stronger metaproperties

 * Strongly join-closed subgroup property: This is a subgroup property that is both join-closed and trivially true
 * Upward-closed subgroup property

Weaker metaproperties

 * Finite-join-closed subgroup property
 * Conjugate-join-closed subgroup property
 * Normal closure-closed subgroup property
 * Characteristic closure-closed subgroup property