Exercise 5, § 1

Let A be the closed subgroup of $\mathbf{G}\mathbf{L}(3,\mathbf{R})$ consisting of the matrices $(a_{ij})$ such that $a_{ij}= 0$ for $i > j,a_{ii}= 1$ for $1\leq i\leq 3,a_{12}\in \mathbf{Z},a_{23}\in \mathbf{Z}$; let $\theta \in$ R -- Q and let B be the subgroup of A consisting of the matrices $(a_{ij})$ such that $a_{12}=a_{23}= 0$ and $a_{13}\in \theta \mathbf{Z}$, and let $G = A/B$.

a) Show that G is a Lie group of dimension one, whose Lie algebra is compact. b) Show that $C(G) =\overline{D(G)}= G_0$, and that $G_0$ is the largest compact subgroup of G.

c) Show that G is not the semi-direct product of $G_0$ with any subgroup.