Lattice theorem

In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence 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 $G$ such that $A$ contains $N$ , 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.

References

W.R. Scott: Group Theory, Prentice Hall, 1964.

Lattice theorem

See also

References