Second-Cohomology Torsion of Finite Abelian Quandles

Definitions and mathematical statement

A quandle, written in the right-operation convention, is a set $X$ with a binary operation $\triangleright$ such that $x\triangleright x=x$, each map $R_y\colon x\mapsto x\triangleright y$ is bijective, and \[ (x\triangleright y)\triangleright z=(x\triangleright z)\triangleright(y\triangleright z) \qquad(x,y,z\in X). \] It is finite when $|X|<\infty$. This convention does not resolve the source's unspecified meaning of "abelian."

For an abelian group $B$, written additively, its torsion subgroup is \[ \operatorname{Tor}(B)=\{b\in B:\text{there exists an integer }n\geq 1\text{ such that }nb=0\}. \] Requested computation, subject to unresolved specifications. For every finite abelian quandle $X$, compute \[ \operatorname{Tor}\left(H^2(X)\right). \] Here $H^2(X)$ retains the source's notation for a second cohomology group. The source record does not specify the cohomology theory, coefficient group, any required coefficient action, or the definition of an abelian quandle. Consequently, the displayed request is a formulation schema rather than a uniquely specified computation. The torsion-subgroup definition above applies when the intended cohomology group is an abelian group; no coefficient choice or interpretation of "abelian" is imposed in this draft.

Short English statement

Determine the torsion in the second cohomology of every finite abelian quandle. The intended meaning of "abelian" and the cohomology theory and coefficients must be specified before this request has a unique mathematical meaning.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004307$, title "Second-Cohomology Torsion of Finite Abelian Quandles," source problem code catalogue entry, collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; 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. This is also the supplied literature reference labelled "original OWR report."
• Supplied literature reference: Lebed--Mortier, "abelian quandles and structure groups." https://doi.org/10.1016/j.jpaa.2020.106474.
• Supplied literature reference: Lebed--Mortier, "detailed preprint." https://arxiv.org/abs/1908.06745.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports that Lebed--Mortier describe the torsion structurally through finite-group quotients and settle important subclasses, while a uniform explicit computation for all finite abelian quandles remains unavailable. This is supplied dated curation, not an independently verified assessment. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.