Exercise 3, § 1

Let G be a discrete subgroup of $\mathbf{R}^n$. For a subgroup H of G to be a direct summand of G it is necessary and sufficient that $H = V \cap G$ where V is a vector subspace of $\mathbf{R}^n$ (for necessity, use Exercise 2; for sufficiency, note that H is also the intersection of G and the vector subspace generated by H).