Modular property of groups

This result is related to the lattice of subgroups in a group

Statement

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