Exercise 5, § 1

For every $a$ belonging to the multiplicative group $\mathbf{Q}^*$, let $\varphi(a)$ be the automorphism of the additive group $\mathbf{R}$ defined by $\varphi(a)x = ax$. Equip $\mathbf{Q}^*$ with the discrete topology. Let G be the topological semi-direct product of $\mathbf{Q}^*$ and $\mathbf{R}$ defined by $\varphi$ (GT, III, §2, No. 10). Show that G is locally isomorphic to $\mathbf{R}$, but is not unimodular.