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 . 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. See also Modular lattice References Isomorphism theorems