Lemma 1

Suppose that $p \neq 0$. Let $G$ be a Lie group, $G_1$ an open subgroup of $G$ which is isomorphic to a standard group and $x \in G$. The following conditions are equivalent: (i) there exists a power of $x$ which belongs to $G_1$; (ii) there exists a strictly increasing sequence $(n_i)$ of integers such that $x^{n_i}$ tends to $e$ as $i$ tends to $+\infty$. (ii) $\Rightarrow$ (i): obvious. (i) $\Rightarrow$ (ii): suppose that $y = x^m \in G_1$. By Proposition 9 (i) of no. 5, $y^{p^n}$ tends to $e$ as $n$ tends to $+\infty$, in other words $x^{m p^n}$ tends to $e$ as $n$ tends to $+\infty$.