Correspondence theorem
In group theory, 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 containing, onto the set of all subgroups of the quotient group. Loosely speaking, 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 and are in then
- if and only if ;
- if then, where is the index of in ;
- where is the subgroup of generated by
- , and
- is a normal subgroup of if and only if is a normal subgroup of.
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. A proof of the correspondence theorem can be found .
Similar results hold for rings, modules, vector spaces, and algebras. More generally an analogous result that concerns congruence relations instead of normal subgroups holds for any algebraic structure.