Minimal Subdegrees of Twisted-Wreath Permutation Groups

Definitions and mathematical statement

Let $G$ be a finite primitive permutation group in the O'Nan--Scott twisted-wreath (TW) class, with the particular decomposition from the source's Theorem 3 \[ G=\operatorname{Soc}(G)\rtimes P\ \leq\ H\wr S_k \] in product action on $\Delta^k$, where $H\leq\operatorname{Sym}(\Delta)$ is the specified primitive component. The socle is the subgroup generated by all minimal normal subgroups; in the TW situation it is a regular nonabelian direct power of a simple group, and a point stabilizer is a complement represented here by $P$. Regular means transitive with trivial stabilizers in the socle action. Primitivity means transitivity with no nontrivial invariant partition.

For a finite transitive group $L$ on $\Lambda$, fix $\alpha\in\Lambda$ and define the minimal subdegree, in the orbit-away-from-the-basepoint convention, by \[ \operatorname{MinSubDeg}(L)= \min_{\beta\in\Lambda\setminus\{\alpha\}}|\beta^{L_\alpha}|. \] It is independent of the basepoint. For the nonregular primitive actions at issue, other point stabilizer orbits are nontrivial; if a different convention excludes all size-one orbits, that convention must be matched to the missing source definitions.

The question, called Question 4 in the supplied assessment, is whether the bound of Theorem 1 is always strict in this TW setting: \[ \operatorname{MinSubDeg}(G) < k\,\operatorname{MinSubDeg}(H). \] Thus the target excludes equality with the stated upper bound, not merely violation of a non-strict estimate. The integer $k$, component $H$, complement $P$ and wreath embedding must be those supplied by Theorem 3. The theorem itself, its restrictions on $k$ and its construction of these data are absent. The quantifier is not silently broadened to arbitrary wreath embeddings or arbitrary choices of $H$; in particular the corrected standalone sentence does not recover all qualifications in the original "$P,H$ as in Theorem 3".

The supplied literature triage checked 21 August 2026 reports bounds, constructions of minimal suborbits and varied twisted-wreath behavior in the primary study, but says the universal strict inequality remains a question there and that no complete resolution was found. These are dated supplied findings, not an independent current-status assessment.

Short English statement

For the source's specified product-action realization of a primitive twisted-wreath group, must its smallest nontrivial suborbit be strictly smaller than $k$ times that of its primitive component? The missing Theorem 3 is needed to specify the permitted decomposition; the supplied assessment leaves the universal strict bound unresolved.

Sources

• Local UnsolvedMath registry, permanent ID 30000203, exact source code catalogue entry; Minimal Subdegrees of Twisted-Wreath Permutation Groups. Proposed year 2005, status open. The original explicitly restricts $P,H$ to Theorem 3 and refers to the bound of Theorem 1; their full statements are not supplied.
• Groups and Geometries (2005), Oberwolfach Reports, DOI 10.4171/owr/2005/12, pp. 647--721, https://doi.org/10.4171/owr/2005/12.
• Supplied literature triage checked 2026-08-21: "original question", https://doi.org/10.4171/owr/2005/12; "primary minimal-subdegree study", https://doi.org/10.1515/9783110199741.75. It identifies this as Question 4 and reports partial bounds and examples without a universal resolution. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.