Kourovka Notebook Problem 21.29

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.

Let $G\leq\operatorname{Sym}(\Omega)$ act on a finite set. Write $G_\alpha=\{g\in G:g\alpha=\alpha\}$ and $G_{\alpha,\beta}=G_\alpha\cap G_\beta$ for point and pointwise two-point stabilizers. The action is primitive if it is transitive and has no invariant partition except the singleton and one-part partitions. A regular suborbit is an orbit of a point stabilizer $G_\alpha$ on which that stabilizer acts freely; equivalently, some $G_{\alpha,\beta}$ is trivial.

Statement recorded in the supplied source.

If $G$ is finite and primitive with a regular suborbit, is it true that \[ \forall\alpha,\beta\in\Omega\ \exists\gamma\in\Omega:\quad G_{\alpha,\gamma}=G_{\beta,\gamma}=\{1\}? \] The source quantifies over all $\alpha,\beta$, including the case $\alpha=\beta$. [S2]

Short English statement

In a finite primitive action that has a trivial two-point stabilizer, can one find a point simultaneously giving trivial stabilizers with any prescribed two points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2538 (KOU-21.29). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 T. C. Burness and M. Giudici, Problem 21.29, 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). This version records a negative answer; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.