Lattice Theorem

In group theory, the correspondence theorem






(also the lattice theorem,W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27. and variously and ambiguously the third and fourth isomorphism 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,
: $N \triangleleft G$, a normal subgroup of ''G'',
: $\mathcal = \$, the set of all subgroups ''A'' of ''G'' that contain ''N'', and
: $\mathcal = \$, the set of all subgroups of ''G''/''N'',

then there is a bijective map $\phi: \mathcal \to \mathcal$ such that
: $\phi(A) = A/N$ for all $A \in \mathcal.$

One further has that if ''A'' and ''B'' are in $\mathcal$ then
* $A \subseteq B$ if and only if $A/N \subseteq B/N$;
* if $A \subseteq B$ then $, B:A, = , B/N:A/N,$, where $, B:A,$ is the index of ''A'' in ''B'' (the number of cosets ''bA'' of ''A'' in ''B'');
* $\langle A,B \rangle / N = \left\langle A/N, B/N \right\rangle,$ where $\langle A, B \rangle$ is the subgroup of $G$ generated by $A\cup B;$
* $(A \cap B)/N = A/N \cap B/N$, and
* $A$ is a normal subgroup of $G$ if and only if $A/N$ 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. A proof of the correspondence theorem can be foun
here

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

See also

* Modular lattice

References

{{reflist

Isomorphism theorems