Exercise 5, § 1

Let $G, G'$ be two closed subgroups of $\mathbf{R}^n$ such that $G' \subset G$, and let $V$ (resp. $V'$) be the largest vector subspace contained in $G$ (resp. $G'$). Show that there is a vector subspace $W$ complementary to $V$ such that $G$ is the direct sum of $V$ and the discrete group $K = W \cap G$, and such that $G'$ is the direct sum of $V \cap G'$ and $K \cap G' = W \cap G'$. (If $U$ is a vector subspace complementary to $V'$, use Exercise 3 to show that the discrete group $U \cap G'$ is the direct sum of $U \cap V \cap G'$ and a discrete group $K'$, and take $W$ to be a vector subspace containing $K'$.) Deduce that:

a) The quotient group $G/G'$ is isomorphic to a product group of the form

where $F$ is a finite abelian group.

b) Every closed subgroup and every Hausdorff quotient group of a group of the form $\mathbf{R}^p \times \mathbf{T}^q \times \mathbf{Z}^r \times F$ ($F$ finite and abelian) are of the same form.