Exercise 3, § 1

Let $\mathbf{R}_d$ be the additive group of real numbers with the discrete manifold structure. Let $\mathbf{R}_d$ operate on the analytic manifold $\mathbf{R}$ by the law of operation $(x, y) \mapsto x + y$. Then $\mathbf{R}_d$ operates transitively on $\mathbf{R}$ but $\mathbf{R}$ is not a Lie homogeneous space of $\mathbf{R}_d$.