Exercise 1, § 7

Let G be a Lie group and $\phi : U \to G$ an exponential mapping such that $\mathbf{Z}U \subset U$ and $\phi(rx) = \phi(x)^r$ for all $x \in U$ and all $r \in \mathbf{Z}$. If $p > 0$, $\phi$ is an analytic isomorphism of U onto $\phi(U)$. (If $\phi(x) = \phi(y)$, then $\phi(p^n x) = \phi(p^n y)$ for all $n \in \mathbf{N}$ and hence $x = y$. Let W be an open neighbourhood of 0 in L(G) such that $\phi^{-1}$ is analytic on $\phi(W)$. For all $s \in \phi(U)$, there exist $n \in \mathbf{N}$ and a neighbourhood V of s in $\phi(U)$ such that $t \in V \Rightarrow t^{p^n} \in \phi(W)$.)