Correspondence theorem (group theory)


In the area of mathematics known as group theory, the correspondence theorem, sometimes referred to as the fourth isomorphism theorem or the lattice theorem, states that if is a normal subgroup of a group, then there exists a bijection from the set of all subgroups of containing, 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.
Specifically, if
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
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.
More generally, there is a monotone Galois connection between the lattice of subgroups of and the lattice of subgroups of : the lower adjoint of a subgroup of is given by and the upper adjoint of a subgroup of is a given by. The associated closure operator on subgroups of is ; the associated kernel operator on subgroups of is the identity.
Similar results hold for rings, modules, vector spaces, and algebras.