Exercise 13, § 9

(a) A multiplication is defined on $\mathbf{R}^3$ as follows:

Show that a solvable real Lie group G is thus obtained such that $DG = \mathbf{R}^2 \times \{0\}$. The centre Z of G is $\{0\} \times \{0\} \times 2\pi \mathbf{Z}$.

(b) For $(x, y, z) \in G$, let $\pi(x, y, z)$ be the mapping

of $\mathbf{R}^2$ into $\mathbf{R}^2$. Show that $\pi$ is a morphism of G onto a Lie subgroup $G'$ of the affine group of $\mathbf{R}^2$, generated by the translations and rotations of $\mathbf{R}^2$. Show that $\mathrm{Ker}\, \pi = \mathbf{Z}$.

(c) We canonically identify $L(G) = T_{(0, 0, 0)}G$ with $\mathbf{R}^3$. Let $(e_1, e_2, e_3)$ be the canonical basis of $\mathbf{R}^3$. Show that $[e_1, e_2] = 0, [e_3, e_1] = e_2, [e_3, e_2] = -e_1$.

(d) Show that, for $c \neq 0$,

Deduce that, for all $u \in L(G)$, $Z \subset \exp(\mathbf{R}u)$. Show that $\exp_G : L(G) \to G$ is neither injective nor surjective.