17.64 (2010)

Say that a group $G$ is an $n$-approximation to the Nottingham group $J = N(p)$ (as defined in 12.24) if $G$ is an infinite pro-$p$ group, and $G/\gamma_n(G)$ is isomorphic to $J/\gamma_n(J)$. Does there exist a function $f(p)$ such that, if $G$ is an $f(p)$-approximation to the Nottingham group, then $\gamma_i(G)/\gamma_{i+1}(G)$ is isomorphic to $\gamma_i(J)/\gamma_{i+1}(J)$ for all $i$?

Cf. 14.56. Note that an affirmative solution to this problem trivially implies the now known fact that $J$ is finitely presented as a pro-$p$ group (if $p > 2$), see 14.55.