Exercise 2, § 10

Suppose that $K = \mathbf{Q}_p$. Let G be a finite-dimensional Lie group. Show that the following conditions are equivalent: (a) there exist $x_1, x_2, \ldots, x_n$ in G such that the subgroup of G generated by $\{x_1, \ldots, x_n\}$ is dense in G; (b) G is generated by a compact subset. (To prove that (b) implies (a), observe that, if $(e_1, \ldots, e_n)$ is a basis of L(G), $(\exp \mathbf{Z}_p e_1)(\exp \mathbf{Z}_p e_2) \ldots (\exp \mathbf{Z}_p e_n)$ is a neighbourhood of e.)