Correspondence theorem (group theory)

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 $N$ is a normal subgroup of a group $G$, then there exists a bijection from the set of all subgroups $A$ of $G$ containing $N$, onto the set of all subgroups of the quotient group $G/N$. 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 a monotone 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, if

G is a group,

N is a normal subgroup of G,

$\mathcal{G}$ is the set of all subgroups A of G such that $N\subseteq A\subseteq G$, and

$\mathcal{N}$ is the set of all subgroups of G/N,

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

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

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

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

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.

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

See also

• Modular lattice