What is the fundamental theorem on discrete groups of Euclidean spaces?

Question

I have been reading the book Using Algebraic Geometry by David A. Cox, John Little, Donal O'Shea for a university project. I am not clear as to what exactly in meant by the phrase "the fundamental theorem on discrete groups of Euclidean spaces" on page 334, shown below.

Answer 1

The result named "fundamental theorem" should be the following:

Proposition: A subgroup $S$ of a Euclidean vector space $V$ is a lattice if and only if $S$ is discrete.

Here a lattice means a subgroup of the form $$ S=\mathbb{Z}w_1\oplus \cdots \oplus \mathbb{Z}w_n $$ with linear independent vectors $w_i\in V$.