Proposition 8

Let $n \in \mathbf{Z}$ and $h_n$ be the mapping $x \mapsto x^n$ of $G$ into $G$. Let $a$ be a non-zero ideal of $A$ contained in $m$, such that $n \notin a$. Then $h_n|G(a)$ is an isomorphism of the analytic manifold $G(a)$ onto the analytic manifold $G(na)$.

By definition of standard groups, $h_n$ is equal on the whole of $G$ to the sum of an integral series with coefficients in $A'$. By § 5, formula (4), this series is of the form

Hence, for $x \in G$, where we write $S(x) = x + \sum_{|\alpha| \geq 2} a_\alpha n^{|\alpha|-2} x^\alpha$. This series $S(x)$ defines an analytic mapping, also denoted by $S$, of $G$ into $G$. By Algebra, Chapter IV, § 6, Proposition 8, there exists an integral series $S'$ in $r$ variables with coefficients in $A'$ such that $S'(S(X)) = S(S'(X)) = X$. Hence $S$ is an isomorphism of the analytic manifold $G$ onto itself and, for every non-zero ideal $b$ of $A$ contained in $m$, $S(G(b)) \subset G(b)$, $S'(G(b)) \subset G(b)$ and therefore As $h_n(y) = n^2 S \left( \frac{1}{n} y \right)$ for $y \in nG$, we see that $h_n|nG(b)$ is an isomorphism of the analytic manifold $nG(b)$ onto the analytic manifold $n^2 G(b)$. But, as $n \notin a$, $|n| > |\lambda|$ for all $\lambda \in a$, hence $n^{-1} a \subset m$ and hence $a$ is of the form $nb$ where $b$ is a non-zero ideal of $A$ contained in $m$.