Kourovka Notebook Problem 15.95

Definitions and mathematical statement

Let $p$ be a prime, let $\Omega$ be a nonempty set, and let $G\leq\operatorname{Sym}(\Omega)$ be a finite $p$-group acting transitively on $\Omega$. Thus the permutation action is faithful, and transitivity implies that $\Omega$ is finite. An element $g\in G$ is fixed-point-free if $g(\omega)\ne\omega$ for every $\omega\in\Omega$. Write $|g|$ for its order and define the exponent by \[ \exp(G)=\operatorname{lcm}\{\,|g|:g\in G\,\}. \] Assume that \[ \forall g\in G,\qquad \left(\forall\omega\in\Omega,\ g(\omega)\ne\omega\right) \Longrightarrow |g|=p. \] Question (source formulation). Must $\exp(G)=p$?

Unresolved boundary qualification. The source does not explicitly exclude the trivial group acting on a singleton. In that case the fixed-point-free hypothesis is vacuous and the exponent is $1$. Consequently, precision requires clarification whether nontriviality is implicit or whether the intended conclusion is instead $\exp(G)\mid p$, equivalently $g^p=1$ for every $g\in G$. For nontrivial finite $p$-groups these conclusions coincide. No repair to the source wording is silently imposed here.

The supplied source states that the intended assertion is true for $p=2,3$, and that under the stated hypothesis the exponent is bounded by a function of $p$ alone. These are reported background claims, not independently verified results in this draft.

Short English statement

The question asks whether a finite transitive permutation $p$-group whose fixed-point-free elements all have order $p$ must itself have exponent $p$. The degree-one boundary case requires clarification. The registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $701183$, source code catalogue entry, title Kourovka Notebook Problem 15.95; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 15.95, 15th issue (2002); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry and background attribute the proposal to E. I. Khukhro, with proposed year 2002. The supplied statement separately begins with the attribution "A. Mann, Ch. Praeger"; both attributions are retained without assigning an unprovided relationship between them.
• A. Hassani, M. Khayaty, E. I. Khukhro, C. E. Praeger, J. Algebra 214, no. 1 (1999), 317--337. The source cites this reference for the assertion that the exponent is bounded in terms of $p$. No article title or link is supplied.