Kourovka Notebook Problem 21.99

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 a finite group $G\leq\operatorname{Sym}(\Omega)$ act transitively on a finite set $\Omega$. For $g\in G$, let $\operatorname{Fix}_\Omega(g)=\{\omega\in\Omega:g\omega=\omega\}$. A derangement has no fixed points, but the requested conclusion also permits elements with at least two fixed points.

Statement recorded in the supplied source.

For every such action and every distinct $\alpha,\beta\in\Omega$, is there a $g\in G$ such that \[ g\alpha=\beta\qquad\text{and}\qquad|\operatorname{Fix}_\Omega(g)|\ne1? \] No primitivity assumption or requirement that $g$ be a derangement is imposed. [S2]

Short English statement

Can every ordered pair of distinct points in a finite transitive action be joined by an element having either zero or at least two fixed points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2608 (KOU-21.99). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 P. Müller, Problem 21.99, 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 refutation; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.