Corollary

Suppose that $K$ is locally compact and let $q = \mathrm{Card}(A/m)$.

(i) If $a = m^a$ and $b = m^b$ with $b \geq a \geq 1$, $G(a)/G(b)$ is a $p$-group of cardinal $q^{r(b-a)}$.

(ii) $G(a)$ is an inverse limit of $p$-groups.

The number of elements in $G(a)/G(b)$ is $(\mathrm{Card}(a/b))^r$; if $b = a + 1$, $a/b$ is a 1-dimensional vector space over $A/m$, whence (i) in this case; the general case follows by induction on $b - a$. Assertion (ii) follows from (i) and Proposition 5 (iii).