# Correspondence theorem (group theory)

In the area of mathematics known as group theory, the correspondence theorem, sometimes referred to as the fourth isomorphism theorem[note 1][note 2] 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.

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.

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

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

## See also

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