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