Kourovka Notebook Problem 21.27

Definitions and mathematical statement

Let $G\le\operatorname{Sym}(\Omega)$ be a finite nonabelian simple group acting transitively on the finite set $\Omega$. Being a permutation subgroup specifies a faithful action: only the identity fixes every point. Transitivity means that for all $\alpha,\beta\in\Omega$ some $g\in G$ satisfies $g\alpha=\beta$. Simplicity means that $G$ has no normal subgroup other than $1$ and itself.

An element $g$ is a derangement in this particular action if it fixes no point of $\Omega$. Write \[ D=\{g\in G:g\alpha\ne\alpha\text{ for every }\alpha\in\Omega\}. \] Does the set product $DD=\{ab:a,b\in D\}$ equal $G$ for every such action? In explicit terms, for every $g\in G$ must there be derangements $a,b$ with $g=ab$?

The two factors are not required to be distinct and may depend on $g$. The identity is included among the target elements; its representation would use inverse derangements. The property is action-dependent, since the same abstract simple group can have different transitive permutation representations and different sets $D$.

Every faithful transitive action of every finite nonabelian simple group is included; primitivity of the action or a restriction on point stabilizers is not assumed. The claim concerns products of exactly two derangements, rather than merely generation of $G$ by derangements or representation by an unbounded number of factors.

The nonabelian convention is explicit here. If cyclic simple groups are included literally, the regular action of $C_2=\{1,a\}$ has $D=\{a\}$ and $DD=\{1\}$, an elementary exception to the printed wording.

Short English statement

In every finite nonabelian simple transitive permutation group, is each element a product of two fixed-point-free permutations from that group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2536 (KOU-21.27), 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 21.27. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.
• S3 M. Larsen, A. Shalev and P. H. Tiep, Products of derangements in simple permutation groups, Forum of Mathematics, Sigma10 (2022). Primary literature for the simple-group derangement problem. https://doi.org/10.1017/fms.2022.69.