Exercise 1, § 1

Let G be a discrete subgroup of rank $p$ in $\mathbf{R}^n$ and let $(\mathbf{a}_i)_{1 \leq i \leq p}$ be a free system of $p$ points of G. The proof of Theorem 1 of no. 1 shows that G is a subgroup of the group generated by the $p$ points $d^{-1} \mathbf{a}_i$, where $d$ is some integer. Show, without using Theorem 1, that there exists a free system of $p$ points $\mathbf{b}_i = \sum_{j=1}^p b_{ij} \mathbf{a}_j$ of G such that, if $x_i = \sum_{j=1}^p x_{ij} \mathbf{a}_i \quad (1 \leq i \leq p)$ is any free system of $p$ points of G, we have $|\det(x_{ij})| \geq |\det(b_{ij})| > 0$. Hence give a proof of Theorem 1 which does not rest on the theory of invariant factors, by showing that the $\mathbf{b}_i$ generate G. (Argue by contradiction: if $z = \sum_{i=1}^p z_i \mathbf{b}_i \in G$ and one of the $z_i$ is not an integer, there exists a point $u = \sum_{i=1}^p u_i \mathbf{b}_i$ of the group generated by $z$ and the $\mathbf{b}_i$ such that $0 < u_i < 1$ for some index $i$, and hence obtain a contradiction.)