Lattice theorem

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In mathematics, the lattice theorem, sometimes improperly referred to as the fourth isomorphism theorem or the correspondence theorem, states that there exists a bijection from the set of all subgroups of a group G that contain a normal subgroup N onto the set of all subgroups of the quotient group G/N. This means that the structure of the subgroups of G/N is exactly the same as the structure of the subgroups of G containing N, with N collapsed to the identity element.

This establishes an antitone Galois connection between the lattice of subgroups of G and the lattice of subgroups of G/N, where the associated closure operator on subgroups of G is $\bar H = HN$ .

Specifically, for a group G and a normal subgroup N of G, there exists a bijection from the set of all subgroups A of G containing N onto the set of subgroups A′ of G/N that maps a subgroup A of G to a subgroup A′ = A/N of G/N. For all A,B ≤ G containing N, and subgroups of G/N A′ = A/N and B′ = B/N, the following hold:

A ≤ B if and only if A′ ≤ B′,
if A ≤ B, then the index of A in B equals the index of A′ in B′,
<A,B>/N = <A′,B′>, where <A,B> is the subgroup of G generated by A ∪ B,
(A ∩ B)/N = (A′) ∩ (B′), and
A is a normal subgroup in G if and only if A′ is a normal subgroup in G′.

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.