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

ϕ(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.

See also

• Modular lattice

References

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