Theorem 1

Every discrete subgroup $G$ of $\mathbf{R}^n$, of rank equal to $p$, is generated by a free system of $p$ points.

By virtue of the properties of groups isomorphic to $\mathbf{Z}^p$ recalled earlier, it is enough to show that $G$ is a subgroup of a discrete group generated by a free system of $p$ points. Now since $G$ is of rank $p$ there exists a free system of $p$ points $a_i$ ($1 \leq i \leq p$) of $G$ such that every $x \in G$ is equal to a linear combination $\sum_{i=1}^p t_i a_i$ of the $a_i$ with real coefficients; since $G$ is discrete, Proposition 1 shows that the $t_i$ are rational. Furthermore, Proposition 1 shows that $G$ is generated by a finite number of points; since these points are linear combinations of the $a_i$ with rational coefficients, there exists an integer $d$ such that they are linear combinations with integer coefficients of the $p$ points $(1/d)a_i = a'_i$. It follows that $G$ is a subgroup of the group generated by the $a'_i$.

Theorem 1 can also be proved without recourse to the theory of invariant factors (cf. Exercise 1).

Discrete subgroups of $\mathbf{R}^n$ which are of rank $n$ are also called lattices in $\mathbf{R}^n$.