Proposition 5

Let $G$ be a standard group.

(i) If $a$ is a non-zero ideal of $A$ contained in $m$, $G(a)$ is an open normal subgroup of $G$.

(ii) The $G(a_\lambda)$, for $\lambda > 0$, form a fundamental system of neighbourhoods of $e$ in $G$.

(iii) Suppose that $a_\lambda \subset a$ for $\lambda \geq \lambda_0$ and let the $G(a)/G(a_\lambda)$, for $\lambda \geq \lambda_0$, be given the discrete topology. Then the topological group $G(a)$ is the inverse limit of the groups $G(a)/G(a_\lambda)$.

(iv) Let $a, b$ be non-zero ideals of $A$ contained in $m$ such that $a \supset b \supset a^2$. The mapping $x \mapsto (x_1 \bmod b, \ldots, x_r \bmod b)$ of $G(a)$ into $(a/b) \times \cdots \times (a/b)$ defines on passing to the quotient an isomorphism of the group $G(a)/G(b)$ onto the additive group $(a/b) \times \cdots \times (a/b)$.

If $x \in G$ and $y \in G(a)$, the coordinates of $x$ and $x.y$ are equal modulo $a$. Hence, for $x', x''$ in $G$ and $y', y''$ in $G(a)$, the coordinates of $x'.x''$ and $(x'.y').(x''.y'')$ are equal modulo $a$. This proves (i).

(ii) is obvious.

(iii) follows from the above and General Topology, Chapter III, § 7, Proposition 2.

If $x \in G(a)$ and $y \in G(a)$, the coordinates of $x.y$ are congruent to those of $x + y$ modulo $G(a^2)$ by formula (4) of § 5. This proves (iv).