# Lattice theorem

In mathematics, the **lattice theorem**, sometimes referred to as the fourth isomorphism theorem or the **correspondence theorem**, states that if is a normal subgroup of a group , then there exists a bijection from the set of all subgroups of such that contains , onto the set of all subgroups of the quotient group . The structure of the subgroups of is exactly the same as the structure of the subgroups of containing with collapsed to the identity element.

This establishes a monotone Galois connection between the lattice of subgroups of and the lattice of subgroups of , where the associated closure operator on subgroups of is

Specifically, if

*G*is a group,*N*is a normal subgroup of*G*,- is the set of all subgroups
*A*of*G*such that , and - is the set of all subgroups of
*G/N*,

then there is a bijective map such that

One further has that if *A* and *B* are in , and *A' = A/N* and *B' = B/N*, then

- if and only if ;
- if then , where is the index of
*A*in*B*(the number of cosets*bA*of*A*in*B*); - where is the subgroup of generated by
- , and
- is a normal subgroup of if and only if is a normal subgroup of .

This list is far from exhaustive. In fact, most properties of subgroups are preserved in their images under the bijection onto subgroups of a quotient group.

Similar results hold for rings, modules, vector spaces, and algebras.

## See also

## References

- W.R. Scott:
*Group Theory*, Prentice Hall, 1964.