Intermediate subgroup condition equals upper intersection-closed

This article gives a proof/explanation of the equivalence of multiple definitions for the term intermediate subgroup condition
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a subgroup property $p$:

• $p$ satisfies the intermediate subgroup condition: whenever a subgroup $H$ of a group $G$ satisfies property $p$ in $G$, $H$ also satisfies property $p$ in any intermediate subgroup $K$.
• $p$ is closed under finite upper intersections: If $H\leq G$ and $K_{1},K_{2}$ are intermediate subgroups of $G$ containing $H$, such that $H$ satisfies property $p$ in both $K_{1}$ and $K_{2}$, then $H$ satisfies property $p$ in $K_{1}\cap K_{2}$.
• $p$ is closed under arbitrary upper intersections.