# Lattice theorem

In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if ${\displaystyle N}$ is a normal subgroup of a group ${\displaystyle G}$, then there exists a bijection from the set of all subgroups ${\displaystyle A}$ of ${\displaystyle G}$ such that ${\displaystyle A}$ contains ${\displaystyle N}$, onto the set of all subgroups of the quotient group ${\displaystyle G/N}$. The structure of the subgroups of ${\displaystyle G/N}$ is exactly the same as the structure of the subgroups of ${\displaystyle G}$ containing ${\displaystyle N,}$ with ${\displaystyle N}$ collapsed to the identity element.

This establishes a monotone Galois connection between the lattice of subgroups of ${\displaystyle G}$ and the lattice of subgroups of ${\displaystyle G/N}$, where the associated closure operator on subgroups of ${\displaystyle G}$ is ${\displaystyle {\bar {H}}=HN.}$

Specifically, if

G is a group,
N is a normal subgroup of G,
${\displaystyle {\mathcal {G}}}$ is the set of all subgroups A of G such that ${\displaystyle N\subseteq A\subseteq G}$, and
${\displaystyle {\mathcal {N}}}$ is the set of all subgroups of G/N,

then there is a bijective map ${\displaystyle \phi :{\mathcal {G}}\to {\mathcal {N}}}$ such that

${\displaystyle \phi (A)=A/N}$ for all ${\displaystyle A\in {\mathcal {G}}.}$

One further has that if A and B are in ${\displaystyle {\mathcal {G}}}$, and A' = A/N and B' = B/N, then

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.

Similar results hold for rings, modules, vector spaces, and algebras.