# Complete Lattice: Lattice of normal subgroups

Complete Lattice: Lattice of normal subgroups
Given two normal subgroups, N and M, of G, their intersection {\displaystyle N\cap M}{\displaystyle N\cap M}and their product {\displaystyle NM=\{nm\mid n\in N\;{\text{ and }}\;m\in M\}}{\displaystyle NM=\{nm\mid n\in N\;{\text{ and }}\;m\in M\}} are also normal subgroups of G.

The normal subgroups of G form a lattice under subset inclusion with least element, {e} , and greatest element, G. The meet of two normal subgroups, N and M, in this lattice is their intersection and the join is their product.

The lattice is complete and modular.[17]