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Hayashi's conjecture on finite connected quandles

Let Q=(Q,⋆)Q=(Q,\star) be a quandle, meaning that a⋆a=aa\star a=a for every a∈Qa\in Q and every left translation La ⁣:Q→QL_a\colon Q\to Q, x↦a⋆xx\mapsto a\star x, is an automorphism. The left multiplication group is LMlt⁡(Q)=⟨La∣a∈Q⟩≤SQ\operatorname{LMlt}(Q)=\langle L_a\mid a\in Q\rangle\leq S_Q; the quandle is connected if this group acts transitively on QQ. A permutation of a finite set has a regular cycle if every cycle length divides the length of its longest cycle, and a quandle has the Hayashi property if all its left translations contain a regular cycle. Hayashi's conjecture. Every finite connected quandle has the Hayashi property. This conjecture, proposed by Hayashi in 2013, concerns the cycle structure of left translations in finite connected quandles; the supplied text does not state whether it has been resolved.

Sources & referencesView supporting material

Primary source

Filip Filipi, “Hayashi Property for Conjugation Quandles”, arXiv:2503.22377 (2025).

finite-groupspermutation-groupsquandles

Progress summary

Refreshed
Partially solved

The conjecture remains open, with progress only for related restricted cases and no verified proof or counterexample.

Proposed by Hayashi in 2013, the conjecture says that every finite connected quandle has the Hayashi property: each left translation contains a regular cycle.

Known results

  • The profile-divisibility formulation is trivial for c=2c=2.
  • Watanabe verified that formulation for c=3c=3 and order n≤47n\leq 47.
  • A 2024 paper proves the profile-divisibility formulation for c∈{3,4,5}c\in\{3,4,5\}.
  • Earlier work proves the profile-divisibility statement in particular cases for finite connected racks.

2025--2026 developments

A 2025 paper restates the supplied left-translation conjecture and explicitly says it remains open for conjugation quandles. The 2026 journal publication revisits the conjecture, but the retrieved record gives no claim of a proof or disproof; no source reports a counterexample, claimed proof, or verification.

Current status (as of August 2026): The supplied left-translation conjecture remains open; only related profile-divisibility cases and special families are settled.

Sources

Solutions 0

No solutions have been posted yet.