Groups Arising as Quandle Structure Groups

Definitions and mathematical statement

Use the following explicit right-quandle convention. A quandle is a set $Q$ with a binary operation $\triangleright:Q\times Q\to Q$ such that, for all $x,y,z\in Q$, \[ x\triangleright x=x,\qquad (x\triangleright y)\triangleright z=(x\triangleright z)\triangleright(y\triangleright z), \] and, for each $y\in Q$, the map $R_y:Q\to Q$, defined by $R_y(x)=x\triangleright y$, is bijective. With this convention, define its structure group by \[ \operatorname{Grp}(Q)=\left\langle e_x\ (x\in Q)\ \middle|\ e_{x\triangleright y}=e_y^{-1}e_xe_y\quad(x,y\in Q)\right\rangle. \] The supplied record does not specify the operation convention or the defining presentation of $\operatorname{Grp}(Q)$; this standard interpretation requires confirmation against the source. It also does not specify whether empty quandles are admitted, so that boundary convention remains unresolved.

The problem is to characterize the class of abstract groups \[ \mathcal C=\left\{G\ \middle|\ G\text{ is a group and there exists a quandle }Q\text{ with }G\cong\operatorname{Grp}(Q)\right\}, \] by giving necessary and sufficient conditions for membership, subject to the convention qualifications above, and to further analyze the groups in this class. No finiteness, cardinality, or connectedness restriction on $Q$, and no finiteness restriction on $G$, is supplied. The phrase "further analyze" specifies no additional mathematical target.

Short English statement

Characterize exactly which abstract groups occur as structure groups of quandles, and investigate their properties. The displayed definition uses an explicit right-quandle convention requiring source confirmation; the empty-quandle boundary and the intended scope of further analysis remain unspecified. The supplied record labels the problem open, without independent verification here.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30004317$, title "Groups Arising as Quandle Structure Groups," source problem code catalogue entry, collection "Oberwolfach Reports Open Problems." No individual proposer is identified; the proposed year is recorded as 2019.
• Supplied report citation: Mini-Workshop: Algebraic Tools for Solving the Yang--Baxter Equation (2020), Oberwolfach Reports, pp. 3207--3242, DOI 10.4171/owr/2019/51. https://doi.org/10.4171/owr/2019/51. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Lebed, Conjugation groups and structure groups of quandles, supplied literature reference: https://arxiv.org/abs/2407.02955.
• Supplied reference labelled "Published version on conjugation and quandle groups": https://doi.org/10.1515/jgth-2025-0096.
• Supplied reference labelled "Oberwolfach Reports, quandle structure groups": https://ems.press/content/serial-article-files/46831.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that the Ryder characterization fails in general, reports computations of structure groups for important quandle families, and states that a general characterization remains open. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.