Permuting upper join-closed subgroup property

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:


 * 1) $$H$$ satisfies $$p$$ in both $$K_1$$ and $$K_2$$, and
 * 2) $$K_1K_2 = K_2K_1$$, i.e., they are defining ingredient::permuting subgroups,

we must have that $$H$$ satisfies $$p$$ in the defining ingredient::join of subgroups $$\langle K_1, K_2 \rangle$$ which in this case is also the defining ingredient::product of subgroups $$K_1K_2$$.

Stronger metaproperties

 * Weaker than::Upper join-closed subgroup property
 * Weaker than::Finite-upper join-closed subgroup property