Strongly join-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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Definition

A subgroup property $p$ is termed strongly join-closed if for any (possibly empty, possibly finite and possibly infinite) collection $H_{i},i\in I$ of subgroups of a group $G$ such that each $H_{i}$ satisfies $p$ in $G$, the join of subgroups $\langle H_{i}\rangle _{i\in I}$ also satisfies $p$ in $G$.

By convention, if $I$ is empty, the join is taken as the trivial subgroup. In particular, a subgroup property is strongly join-closed if and only if it is join-closed and trivially true (i.e., always satisfied by the trivial subgroup).

Relation with other metaproperties

Stronger metaproperties

Metaproperty name	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Strongly UL-join-closed subgroup property

Weaker metaproperties

Metaproperty name	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
join-closed subgroup property
finite-join-closed subgroup property
strongly finite-join-closed subgroup property