Modular property of groups

Symbolic statement
Let $$A$$, $$B$$ and $$C$$ be subgroups of a group $$G$$ with the property that $$A \le C$$. Then:

$$A (B \cap C) = AB \cap C$$

Note here that $$AB$$ denotes the product of subgroups and is not in general a group.

Implications
In case $$A$$ commutes with the groups $$B$$ and $$B \cap C$$, then the above can be recast as saying that the modular identity holds for the lattice of subgroups. This has the following easy implications:


 * Any permutable subgroup is modular
 * Any normal subgroup is modular