 Solution

We next bound the closed derived series. For odd $p$, the depth calculation yieldsFor the inclusion from left to right, two series of depth at least $m$ commute modulo ${\mathcal{N}}_{2m+1}$: at depth $2m$ the only possible leading pair has equal depths and its coefficient vanishes. For the reverse inclusion, every depth $k\ge 2m+1$ can be written $r+s=k$ with $r,s\ge m$ and $r\not\equiv s(\mathrm{mod}p)$. Start with $r=m,s=k-m$; if their difference vanishes modulo $p$, replace them by $r=m+1,s=k-m-1$, which is allowed except at $k=2m+1$, where the original difference is already nonzero. The difference changes by $2$, nonzero for odd $p$. Commutators thus supply every high leading coefficient, and closed successive coefficient correction fills the tail.

With ${\mathcal{N}}^{(0)}={\mathcal{N}}_1$, induction on $n$ givesFor $p=2$ use the standard characteristic-two derived-series collection bound, explicitly stated as an additional result:for a fixed constant $C$. One can take a fixed enlarged constant from the tail estimate ${\mathcal{N}}^{(n)}\supseteq {\mathcal{N}}_{\lceil 4{\kappa }^n\rceil +1}$. The more involved two-coefficient collection is necessary in characteristic two; the exact odd-prime tail formula is not asserted there.

In either case there are constants $C^{\prime }>0$ and $\lambda <3$ such that ${\log }_p[\mathcal{N}:{\mathcal{N}}^{(n)}]\le C^{\prime }{\lambda }^n$. If an analytic structure existed, its dimension $d$ would be positive, since a zero-dimensional analytic pro-p group is discrete and compact, hence finite. The second alternative of the supplied theorem would requirefor all sufficiently large $n$, for some fixed $s$. Combining the bounds gives $d{3}^{-s}\le C^{\prime }(\lambda /3{)}^n$, impossible as $n\to \infty$.

The root solution proved hereditary just infiniteness, so the supplied theorem applies. Part (i) excluded its linear alternative, and this calculation excludes its growth alternative. Hence the Nottingham group is not analytic over any pro-$p$ ring.