# Modular property of groups

From Groupprops

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

## Statement

### Symbolic statement

Let , and be subgroups of a group with the property that . Then:

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

## Implications

In case *commutes* with the groups and , 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