On a Conjecture by Hayashi on Finite Connected Quandles

A quandle is an algebraic structure whose binary operation is idempotent, right-invertible and right self-distributive. Right-invertibility ensures right translations are permutations and right self-distributivity ensures further they are automorphisms. For finite connected quandles, all right translations have the same cycle structure, called the profile of the connected quandle. Hayashi conjectured that the longest length in the profile of a finite connected quandle is a multiple of the remaining lengths. We prove that this conjecture is true for profiles with at most five lengths.