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
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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. $K_{1}K_{2}=K_{2}K_{1}$, i.e., they are permuting subgroups,

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

Relation with other metaproperties

Stronger metaproperties

• Upper join-closed subgroup property
• Finite-upper join-closed subgroup property