Goldman-Type Theorem for Sharply Transitive Permutation Groups

Definitions and mathematical statement

A factor is a von Neumann algebra with center consisting of scalar multiples of its identity, and a subfactor inclusion is a unital inclusion of factors. Let $\Omega$ be a finite set. For an integer $k$ with $1\leq k\leq |\Omega|$, a permutation group $G\leq\operatorname{Sym}(\Omega)$ is sharply $k$-transitive if, for any two ordered tuples $(x_1,\ldots,x_k)$ and $(y_1,\ldots,y_k)$, each having pairwise distinct entries in $\Omega$, there is exactly one $g\in G$ such that $g x_i=y_i$ for every $i$. Fix $\omega\in\Omega$ and put \[ H=G_\omega=\{g\in G:g\omega=\omega\}. \] An action $\alpha:G\to\operatorname{Aut}(R)$ on a factor $R$ is outer if $\alpha_g$ is not an inner automorphism for every $g\neq e$. Its restriction to $H$ gives the crossed-product inclusion \[ R\rtimes_{\alpha|_H}H\subset R\rtimes_\alpha G. \] In the usual finite-index subfactor setting, the principal graph is the rooted bipartite graph recording the multiplicities arising from the inclusion's iterated bimodule decompositions. The supplied record denotes the principal graph of a reference group--subgroup crossed-product inclusion by $\Gamma_{G>H}$. It does not specify the reference factor, reference action, factor type, or precise principal-graph conventions; these data must be fixed in the intended setting before the assertion is fully determined.

The proposed assertion is the following. For a sharply $k$-transitive finite permutation group $G$ and its point stabilizer $H$, every subfactor inclusion $Q\subset P$ whose principal graph is isomorphic to $\Gamma_{G>H}$ is isomorphic, as an inclusion, to a crossed-product inclusion for this same pair $(G,H)$. Explicitly, there should exist a factor $R$, an outer action $\alpha:G\to\operatorname{Aut}(R)$, and a von Neumann algebra isomorphism \[ \Phi:P\longrightarrow R\rtimes_\alpha G \quad\text{such that}\quad \Phi(Q)=R\rtimes_{\alpha|_H}H. \] The source does not specify the intended range of $k$, beyond using sharply $k$-transitive groups, or the ambient class of subfactors quantified over. No factor type, irreducibility condition, or additional analytic hypothesis is imposed here to fill that gap. The hypothesis names a principal graph only; the supplied statement does not additionally require a dual principal graph or an identification of standard invariants.

The supplied excerpt asserts that the conjecture holds for $k=2,3,4$. This is retained as a source claim, not as an independently verified theorem. Its background discussion also asserts that sharply $3$-transitive permutation groups form exactly two families, denoted $L(q)$ and $M(q)$, which are transitive extensions of groups denoted $H(q)$ and $S(q)$, respectively. It identifies $L(q)=\operatorname{PGL}_2(q)$, describes its fractional-linear action on the projective line, and describes $M(q)$ as having the same underlying set as $\operatorname{PGL}_2(q)$, with an action whose branches depend on whether the determinant is a square or a nonsquare. The admissible parameters, the automorphism denoted $\sigma$, and parts of the displayed formulas are missing or corrupted. The claimed example printed as $S(32)$ and $Z23\rtimes Q_8$ is likewise not unambiguously recoverable. These background assertions therefore do not supply additional precise hypotheses or constructions for the conjecture.

Short English statement

The conjecture asks whether the principal graph associated with a sharply $k$-transitive finite group and a point stabilizer forces every subfactor with that graph to arise from an outer action of the same group and subgroup. The supplied source claims the cases $k=2,3,4$ and labels the general problem open. The intended range of $k$, analytic setting, and some background notation require clarification.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30006362$, exact source problem code catalogue entry, title "Goldman-Type Theorem for Sharply Transitive Permutation Groups." The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2025$; no individual proposer is supplied.
• Supplied report citation: Subfactors and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/34, pp. 1823--1886. Supplied source URL: https://doi.org/10.4171/owr/2025/34. The title year $2026$ and report designation $34/2025$ are retained as supplied.
• Supplied literature source: "Oberwolfach Report 34/2025," https://ems.press/content/serial-article-files/52239.
• Literature assessment dated exactly 2026-08-22: the record labels the problem open and states that only the cases $k=2,3,4$ are established in the record, leaving the general statement unresolved. This is supplied dated curation, not independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The source record labels its statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and notation repaired. These are supplied verification claims; this draft does not claim to have visited or checked the report.
• The original excerpt cites "[4, Chapter XI, Section 2]" for its background on sharply $3$-transitive groups. The bibliographic identity of reference [4] is not supplied.