Kourovka Notebook Problem 21.6

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A permutation of a set $\Omega$ is finitary if it moves only finitely many points; $\operatorname{FSym}(\Omega)$ is the group of these permutations. A $p$-group means that every element has order a power of the prime $p$, and a Sylow $p$-subgroup is a maximal $p$-subgroup. An action is transitive if it has one orbit. A nonempty subset $B\subset\Omega$ is a block for $H$ if $hB=B$ or $hB\cap B=\varnothing$ for every $h\in H$. Following Notebook 15.23, a transitive group is totally imprimitive when every finite subset of $\Omega$ is contained in a finite block. Such a finitary $p$-group has the cyclic-block property if the support of each cycle in the disjoint-cycle decomposition of every member is a block for the entire group.

Statement recorded in the supplied source.

For every prime $p$, set $\Omega$, transitive subgroup $G\leq\operatorname{FSym}(\Omega)$, and transitive Sylow $p$-subgroup $P\leq G$, does there exist a transitive subgroup $H\leq P$ that is totally imprimitive and has the cyclic-block property? [S2]

Short English statement

Does every transitive Sylow p-subgroup of a transitive finitary permutation group contain a transitive subgroup whose individual cycle supports are blocks?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2515 (KOU-21.6). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. O. Asar, Problem 21.6, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). See also source definitions in Problems 15.23. https://arxiv.org/abs/1401.0300v48.