Finite-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 a finite-join-closed subgroup property if, for any group $G$ and subgroups $H$ and $K$ of $G$ such that both $H$ and $K$ satisfy $p$ in $G$, we also have that the join of subgroups $\langle H,K\rangle$ satisfies $p$ in $G$.

Relation with other metaproperties

Stronger metaproperties

• Strongly join-closed subgroup property
• Join-closed subgroup property
• Strongly finite-join-closed subgroup property

Weaker metaproperties

• Permuting join-closed subgroup property
• Normalizing join-closed subgroup property