Jordan–Zassenhaus theorem

Remarks

The proof combines a reduction theory for lattices with the finiteness of the class number. In outline, one fixes a lattice $M_0$ with $K \otimes_R M_0 \cong V$ and finds a nonzero $c \in R$ such that every lattice $M$ with $K \otimes_R M \cong V$ is isomorphic to a lattice $N$ with $c M_0 \subseteq N \subseteq M_0$. There are only finitely many such $N$, because $M_0 / c M_0$ is finite. Example: take $A = K$, $\Lambda = R$ and $V = K$. The $\Lambda$-lattices with $K M \cong K$ are the nonzero fractional ideals of $R$, up to isomorphism, so in this case the theorem is the finiteness of the class number of $K$. Example: take $K = \Q$ and $\Lambda = \Z G$ for a finite group $G$; there are finitely many $\Z G$-lattices of each $\Z$-rank $n$ up to isomorphism. Together with Minkowski's bound on the orders of finite subgroups of $GL_n(\Z)$, it follows that $GL_n(\Z)$ has only finitely many conjugacy classes of finite subgroups.

Going deeper

For a student

The theorem is the basic finiteness result of integral representation theory. It lets one define the class number of an order, the number of isomorphism classes of lattices in a given genus. This number can then be studied with adelic methods. It extends to any Dedekind domain whose field of fractions is a global field, with $A$ a separable algebra. Applied to integral group rings, it controls the integral representations of finite groups, and in crystallography it gives the finiteness of arithmetic classes of finite subgroups of $GL_n(\Z)$.