Diameter Bounds with Permutation-Rich Generating Sets

Definitions and mathematical statement

Let $F$ be a finite field, let $n$ be a positive integer, and write \[ H=\operatorname{SL}_n(F)=\{g\in\operatorname{Mat}_{n\times n}(F):\det g=1\}. \] For $\sigma\in\operatorname{Sym}(n)$, let $P_\sigma$ be its permutation matrix, defined by $P_\sigma e_i=e_{\sigma(i)}$. The supplied hypothesis is that $A\subseteq H$, $\langle A\rangle=H$, and $P_\sigma\in A$ for every $\sigma\in\operatorname{Sym}(n)$.

There is an unresolved compatibility issue in this hypothesis: $\det P_\sigma=\operatorname{sgn}(\sigma)$ in $F$. For $n\geq2$ in odd characteristic, odd permutation matrices do not belong to $H$, so no such $A$ exists. In characteristic two, all permutation matrices do belong to $H$. The record does not specify whether characteristic two was intended, whether only even permutation matrices were intended, or whether a different embedding or ambient group was intended. These alternatives are not interchangeable and are not silently substituted here.

The notation $\Gamma(H,A)$ also leaves the graph convention unspecified. To expose this issue, define the undirected word diameter by \[ D_{\pm}(H,A)=\max_{h\in H}\min\{\ell\geq0:h=a_1\cdots a_\ell,\ a_i\in A\cup A^{-1}\}, \] and define the directed word diameter by \[ D_+(H,A)=\max_{h\in H}\min\{\ell\geq0:h=a_1\cdots a_\ell,\ a_i\in A\}. \] The empty product is the identity. Since $H$ is finite and $A$ generates $H$, both quantities are finite. They are the diameters of, respectively, the undirected Cayley graph with edges $\{x,xa\}$ and the directed Cayley graph with arcs $x\to xa$, for $x\in H$ and $a\in A$.

Problem, with unresolved qualifications. Establish a strong upper bound for the intended diameter of $\Gamma(H,A)$ under the supplied hypotheses, after resolving the permutation-matrix and graph-convention issues. The source tentatively suggests a polylogarithmic bound. A precise uniform version of that suggested target is: there exist absolute constants $C,c>0$ such that, for every admissible $n,F,A$, \[ \operatorname{diam}\Gamma(H,A)\leq C\left(\log|H|\right)^c. \] Here $C$ and $c$ would be independent of $n$, $F$, and $A$; the logarithm may be taken to be natural. The trivial case $|H|=1$ has diameter zero. This display formalizes the suggested ideal target, not an established bound or a fully specified conjecture in the source: the original request is for a "good bound" and presents the polylogarithmic expression as a possibility.

The original excerpt additionally suggests assuming only that $A$ contains a couple of permutation matrices generating $\operatorname{Alt}(n)$ or $\operatorname{Sym}(n)$, described there as subgroups of $\operatorname{SL}_n$, and then applying an unidentified result labelled $(2)$. The symmetric-group version has the same determinant obstruction for the standard permutation representation in odd characteristic. The excerpt supplies neither result $(2)$ nor sufficient details to formalize this proposed reduction. Its surrounding discussion of flag stabilizers and relatively large subsets, written as $A>|F|$, does not clearly impose an additional cardinality hypothesis on this problem.

Short English statement

For a finite special linear group generated by a set containing all permutation matrices, seek a strong Cayley graph diameter bound, ideally a power of the logarithm of the group order with absolute constants. The literal hypothesis is impossible in odd characteristic when $n\geq2$, and the intended permutation subgroup and directed or undirected graph convention require clarification. The supplied open status and literature assessment are dated curation, not independently verified facts.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json designated as the supplementary registry; supplied complete record for permanent UnsolvedMath ID $30005879$, titled Diameter Bounds with Permutation-Rich Generating Sets. Exact source problem code: catalogue entry; the extracted original text labels it "Problem 1." The supplied attribution is to the Oberwolfach Reports Open Problems collection, with proposed year 2024; no individual proposer is named.
• Supplied report citation: Mini-Workshop: Growth and Expansion in Groups (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/17, pp. 1005--1040. https://doi.org/10.4171/owr/2024/17. This is the supplied report-level provenance; the record gives no narrower page reference for the problem.
• Supplied literature source, labelled "classical-group diameter bounds": https://arxiv.org/abs/2110.02942. No title, authors, or specific theorem from this reference are supplied.
• Supplied literature source, labelled "OWR report": https://oa.tib.eu/renate/server/api/core/bitstreams/5c03f705-7680-4cb6-b7cd-653d4b29494d/content.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that known classical-group diameter bounds remain rank-dependent or require additional generator hypotheses, and that the suggested polylogarithmic bound with an absolute exponent for this permutation-rich generating set is not established. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified, with an accompanying assertion that only formula delimiters, notation, grammar, and typography were repaired while preserving mathematical content and scope. This records the supplied editorial qualification and does not establish mathematical correctness or resolve the formulation issues above.