Kourovka Notebook Problem 14.94

Definitions and mathematical statement

Let $p$ be a prime and write ${\mathbb{F}}_p={\mathbb{Z}}/p{\mathbb{Z}}$. Interpret $N({\mathbb{F}}_p)$ as the Nottingham group \[ N({\mathbb{F}}_p)=\left\{x+\sum_{n=2}^{\infty}b_nx^n:b_n\in{\mathbb{F}}_p\right\}, \] with formal composition as its group operation and the $x$-adic topology. Thus a neighbourhood basis of the identity consists of the subgroups of series congruent to $x$ modulo $x^m$, for $m\geq2$. For each positive integer $r$, let \[ H_r=\left\{x\left(1+\sum_{i=1}^{\infty}a_i x^{p^r i}\right):a_i\in{\mathbb{F}}_p\right\}\subseteq N({\mathbb{F}}_p), \] which the supplied source describes as a closed subgroup, equipped with the induced topology.

For a profinite group $G$, its $p$-cohomological dimension $\mathrm{cd}_p(G)$ is the least integer $d\geq0$ such that \[ H^n_{\mathrm{cts}}(G,M)=0 \quad\text{for every }n>d\text{ and every discrete }p\text{-primary torsion }G\text{-module }M \] with continuous action; set $\mathrm{cd}_p(G)=\infty$ if no such $d$ exists.

Problem (I. B. Fesenko, Kourovka Notebook 14.94). Determine \[ \mathrm{cd}_p(H_r) \qquad\text{for each integer }r\geq1. \] The source does not explicitly specify or quantify the prime $p$, nor define $N({\mathbb{F}}_p)$. The formulation above uses the standard Nottingham-group interpretation and treats $p$ as an arbitrary fixed prime; any intended restriction on $p$ requires source confirmation. Malformed braces in the supplied coefficient-field notation have been repaired to ${\mathbb{Z}}/p{\mathbb{Z}}$. The supplied registry labels the problem open; that label is not an independently verified mathematical assessment.

Short English statement

For each positive integer $r$, determine the continuous $p$-cohomological dimension of the closed subgroup of the Nottingham group over ${\mathbb{F}}_p$ consisting of series $x(1+\sum_{i\geq1}a_i x^{p^r i})$. The prime range and the standard interpretation of the ambient group require source confirmation.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701078$, exact source problem code catalogue entry, title Kourovka Notebook Problem 14.94; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 14.94, attributed in the supplied record to I. B. Fesenko, proposed in 1999 and associated with the 14th issue (1999).
• The supplied background identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently verified.