Correspondence theorem (group theory)

In the area of mathematics known as group theory, the correspondence theorem,[1][2][3][4][5][6][7][8] sometimes referred to as the fourth isomorphism theorem[6][9][note 1][note 2] or the lattice theorem,[10] 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 ${\displaystyle G}$ containing ${\displaystyle N}$, 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.

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

• ${\displaystyle A\subseteq B}$ if and only if ${\displaystyle A'\subseteq B'}$;
• if ${\displaystyle A\subseteq B}$ then ${\displaystyle |B:A|=|B':A'|}$, where ${\displaystyle |B:A|}$ is the index of A in B (the number of cosets bA of A in B);
• ${\displaystyle \langle A,B\rangle /N=\langle A',B'\rangle ,}$ where ${\displaystyle \langle A,B\rangle }$ is the subgroup of ${\displaystyle G}$ generated by ${\displaystyle A\cup B;}$
• ${\displaystyle (A\cap B)/N=A'\cap B'}$, and
• ${\displaystyle A}$ is a normal subgroup of ${\displaystyle G}$ if and only if ${\displaystyle A'}$ is a normal subgroup of ${\displaystyle G/N}$.

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 ${\displaystyle (f^{*},f_{*})}$ between the lattice of subgroups of ${\displaystyle G}$ (not necessarily containing ${\displaystyle N}$) and the lattice of subgroups of ${\displaystyle G/N}$: the lower adjoint of a subgroup ${\displaystyle H}$ of ${\displaystyle G}$ is given by ${\displaystyle f^{*}(H)=HN/N}$ and the upper adjoint of a subgroup ${\displaystyle K/N}$ of ${\displaystyle G/N}$ is a given by ${\displaystyle f_{*}(K/N)=K}$. The associated closure operator on subgroups of ${\displaystyle G}$ is ${\displaystyle {\bar {H}}=HN}$; the associated kernel operator on subgroups of ${\displaystyle G/N}$ is the identity.

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