Hayashi Property for Conjugation Quandles
Filip Filipi
TL;DR
The paper investigates Hayashi's property for conjugation quandles, reframing Hayashi's conjecture as a centrality condition in the group generated by a finite conjugacy class. It introduces the notion of a good group and proves that if $G$ is good, every finite conjugation quandle $\mathrm{Cl}_G(e)$ enjoys the Hayashi property, thereby validating Hayashi's conjecture for all connected conjugation quandles over such groups. The core technique links quandle dynamics (regular cycles of left translations) to group centralizers via $H=\langle C\rangle$ and the isomorphism $H/Z(H) \cong \mathrm{Inn}(H)|_C$. Using this framework, the authors establish Hayashi-property results for finite nilpotent groups, the symmetric and alternating groups $\mathrm{S}_n$, $\mathrm{A}_n$, and dihedral groups $\mathrm{D}_{2n}$, and report extensive computational verifications up to order 500 and for many simple groups up to order $3.5\cdot 10^6$, reinforcing the conjectural link between centrality in groups and Hayashi’s conjecture for conjugation quandles.
Abstract
We give a comprehensive description of conjugation quandles and their connectedness. In this context, we find a characterization of Hayashi's conjecture (2013) in terms of a centrality condition of groups. This condition is thus a conjecture itself and it states that powers of elements of a finite and generating conjugacy classes should be central whenever they commute with one particular element of this class. We prove this condition in several cases, e.g. for finite nilpotent, symmetric, alternating, and dihedral groups. All of these results translate to Hayashi's conjecture for the corresponding conjugation quandles.