Lattice theorem

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 N {displaystyle N} is a normal subgroup of a group G {displaystyle G} , then there exists a bijection from the set of all subgroups A {displaystyle A} of G {displaystyle G} containing N {displaystyle N} , onto the set of all subgroups of the quotient group G / N {displaystyle G/N} . The structure of the subgroups of G / N {displaystyle G/N} is exactly the same as the structure of the subgroups of G {displaystyle G} containing N {displaystyle N} , with N {displaystyle N} collapsed to the identity element. 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 N {displaystyle N} is a normal subgroup of a group G {displaystyle G} , then there exists a bijection from the set of all subgroups A {displaystyle A} of G {displaystyle G} containing N {displaystyle N} , onto the set of all subgroups of the quotient group G / N {displaystyle G/N} . The structure of the subgroups of G / N {displaystyle G/N} is exactly the same as the structure of the subgroups of G {displaystyle G} containing N {displaystyle N} , with N {displaystyle N} collapsed to the identity element.