# Intermediate subgroup condition equals upper intersection-closed

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 $H \le G$ 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.