Permuting upper 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 upper join-closed subgroup property if for any subgroup $H$ of a group $G$ and two intermediate subgroups $K_{1}$ and $K_{2}$ satisfying:

$H$ satisfies $p$ in both $K_{1}$ and $K_{2}$ , and
$K_{1} K_{2} = K_{2} K_{1}$ , i.e., they are permuting subgroups,

we must have that $H$ satisfies $p$ in the join of subgroups $⟨ K_{1}, K_{2} ⟩$ which in this case is also the product of subgroups $K_{1} K_{2}$ .

Relation with other metaproperties

Stronger metaproperties