Kourovka Notebook Problem 14.93

Definitions and mathematical statement

Let $p$ be a prime and let ${\mathbb{F}}_p={\mathbb{Z}}/p{\mathbb{Z}}$. Under the standard interpretation of the Nottingham group, put \[ N({\mathbb{F}}_p)=\left\{f(x)=x+\sum_{n\geq 2}a_nx^n : a_n\in{\mathbb{F}}_p\right\}, \] with group operation $(f\circ g)(x)=f(g(x))$ and identity $x$. Give this group the topology whose neighbourhoods of the identity are \[ U_m=\{f\in N({\mathbb{F}}_p):f(x)\equiv x\pmod{x^{m+1}}\},\qquad m\geq 1. \] This is the usual pro-$p$ topology. The supplied statement instead refers to Archive Problem 12.24 for its definition; that referenced text is absent from the supplied record, so the identification with this standard definition requires source review.

Problem. Find defining relations for $N({\mathbb{F}}_p)$ as a pro-$p$ group. More precisely, find a topological generating set $S$ and a set $R$ of elements of the free pro-$p$ group $F_{\mathrm{pro}\text{-}p}(S)$ such that the natural continuous homomorphism onto $N({\mathbb{F}}_p)$ has kernel equal to the closed normal subgroup generated by $R$. Thus the requested presentation is \[ N({\mathbb{F}}_p)\cong F_{\mathrm{pro}\text{-}p}(S)/\overline{\langle\!\langle R\rangle\!\rangle}. \] No finiteness requirement on $R$ is stated in the source.

The source asserts that the group has two generators, for example \[ u(x)=x+x^2,\qquad v(x)=\frac{x}{1-x}=\sum_{n\geq 1}x^n. \] Here the fraction denotes a formal power series. In the asserted two-generator formulation, let $F=F_{\mathrm{pro}\text{-}p}(a,b)$ and let $\pi:F\to N({\mathbb{F}}_p)$ be the continuous homomorphism with $\pi(a)=u$ and $\pi(b)=v$. Subject to the source's assertion that these elements topologically generate the group, the task is to describe $R\subseteq F$ with \[ \ker\pi=\overline{\langle\!\langle R\rangle\!\rangle}_F. \] The supplied wording does not restrict the prime in its generator assertion; its validity and intended prime range require verification, including at $p=2$.

Supplied qualification. A comment dated 2009 directs the reader, for $p>2$, to progress by M. V. Ershov in the 2005 reference below. The record does not describe the results or their effect on the remaining task.

Short English statement

Find a presentation of the Nottingham group over ${\mathbb{F}}_p$ using relations in a free pro-$p$ group, with the relation subgroup taken to be closed. The source proposes the generators $x+x^2$ and $x/(1-x)$, but their stated prime range needs review. Its 2009 comment cites progress for $p>2$ without specifying the results.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701077$, exact source code catalogue entry, title Kourovka Notebook Problem 14.93. The registry labels the problem open; this is a supplied status, not an independently verified mathematical assessment. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original Problem 14.93, 14th issue (1999), proposed by I. B. Fesenko. The supplied background identifies the 21st issue edition (2026), arXiv:1401.0300v48; the collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Archive Problem 12.24 of the notebook is the supplied cross-reference for the definition of the Nottingham group, also called the "Wild group". Its text was not supplied.
• M. V. Ershov, J. London Math. Soc. 71 (2005), 362--378. The source's comment dated 2009 cites this reference as progress for $p>2$. No article title or account of its results was supplied. These bibliographic and progress statements have not been independently verified.