Proposition 13

Suppose that $p \neq 0$. If $G$ is compact or standard, then $G_f = G$.

If G is standard, it suffices to use Lemma 1. Suppose that G is compact. Let $x \in G$ and V be a neighbourhood of e in G. Let y be a limit point of the sequence $(x^n)_{n \geq 0}$. For all $n > 0$, there exist two integers $n_1, n_2$ such that $n_1 \geq 2n_2 \geq n$ and $x^{n_1} \in yV, x^{n_2} \in yV$, whence $x^{n_1 - n_2} \in V^{-1}V$ and $n_1 - n_2 \geq n$. Hence $x \in G_f$.