Kourovka Notebook Problem 17.103

Definitions and mathematical statement

Let $\mathcal P$ denote the set of all primes. For $\pi\subseteq\mathcal P$, a finite group is a $\pi$-group if every prime divisor of its order belongs to $\pi$; the trivial group is included. A subgroup $H\leq G$ of a finite group is a Hall $\pi$-subgroup if every prime divisor of $|H|$ belongs to $\pi$ and no prime divisor of $[G:H]$ belongs to $\pi$.

Using the standard meaning of $D_\pi$, a finite group $G$ is a $D_\pi$-group if it has a Hall $\pi$-subgroup, any two Hall $\pi$-subgroups of $G$ are conjugate in $G$, and every $\pi$-subgroup of $G$ is contained in a Hall $\pi$-subgroup of $G$.

Define \[ \mathcal S=\left\{\pi\subseteq\mathcal P:\ \text{for every finite group }G,\quad \left(G\text{ has a Hall }\pi\text{-subgroup}\right) \Longrightarrow \left(G\text{ is a }D_\pi\text{-group}\right) \right\}. \] The question proposed by D. O. Revin in 2010 is whether \[ |\mathcal S|=2^{\aleph_0}. \] Here a continuum of sets means continuum many distinct subsets of $\mathcal P$.

The supplied Comment of 2025 reports the following established special case, attributed to K. A. Ilenko and N. V. Maslova: for every integer $x\in{\mathbb{Z}}$, the set \[ \pi_x=\{p\in\mathcal P:p>x\} \] belongs to $\mathcal S$. This reported result and the registry's open status are supplied source assessments, not independently verified here. The source does not explicitly define $D_\pi$; the formulation above uses its standard group-theoretic definition and requires review against the cited notebook.

Short English statement

Are there continuum many sets $\pi$ of primes such that every finite group with a Hall $\pi$-subgroup satisfies the full Sylow-type existence, conjugacy, and containment properties for $\pi$-subgroups? The supplied 2025 comment reports that every set of primes greater than an integer threshold has this property.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701405$, exact source code catalogue entry, title Kourovka Notebook Problem 17.103. The supplied status is open; the record was created and updated on 2026-10-16T14:00:00Z. These are registry metadata, not an independent mathematical or literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 17.103, proposed by D. O. Revin in 2010. The supplied background identifies the 17th issue (2010) and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These source details have not been independently checked.
• The supplied Comment of 2025 attributes the integer-threshold result to K. A. Ilenko and N. V. Maslova, Siberian Math. J., volume 62, no. 1 (2021), pp. 44--51. No article title or direct link is supplied; the result is retained as a source-reported qualification.