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