Kourovka Notebook Problem 21.28

Definitions and mathematical statement

Let $G\le\operatorname{Sym}(\Omega)$ be a finite nontrivial simple permutation group acting transitively on a finite set $\Omega$. The action is faithful because $G$ is a subgroup of the full permutation group. An element $g\in G$ is a derangement if it has no fixed point, that is, $g\alpha\ne\alpha$ for all $\alpha\in\Omega$. Define the proportion of derangements in this action by \[ \delta(G,\Omega)=\frac{|\{g\in G:g\text{ fixes no point of }\Omega\}|}{|G|}. \] The denominator counts all elements, including the identity, rather than conjugacy classes. A uniformly random group element is therefore a derangement with probability $\delta(G,\Omega)$.

The proposed inequality is \[ \delta(G,\Omega)\ge\frac{89}{325} \] for every such simple group and every faithful transitive action. The source writes $\delta(G)$, with the action understood; displaying $\Omega$ makes this dependence explicit. Simple means nontrivial with no proper nontrivial normal subgroup, and thus includes cyclic prime-order cases as well as nonabelian cases.

The lower bound is one absolute rational number, independent of group order and permutation degree. The claim does not require the action to be primitive, nor impose any structural hypothesis on a point stabilizer. One cannot check only a group's natural representation: all eligible transitive representations are part of the question. A counterexample would be an explicit action with strictly smaller derangement proportion.

Short English statement

Is the fraction of derangements at least $89/325$ in every faithful transitive action of a finite simple group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2537 (KOU-21.28), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.28. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.