Permuting 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 permuting join-closed subgroup property if, for any group $G$ and permuting subgroups $H$ and $K$ of $G$ such that $H$ and $K$ both satisfy $p$ in $G$, we must have that $HK$ 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
• Finite-join-closed subgroup property

Weaker metaproperties

• Normalizing join-closed subgroup property