Correspondence theorem

In group theory, the correspondence theorem (also the lattice theorem, and variously and ambiguously the third and fourth isomorphism 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 G is a group, a normal subgroup of G, the set of all subgroups A of G that contain N, and 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 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. More generally, there is a monotone Galois connection between the lattice of subgroups of (not necessarily containing ) 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 here. 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.