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.

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

See also

• Modular lattice

References

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