On a conjecture about profiles of finite connected racks

TL;DR

This paper investigates the profile of finite connected racks and quandle structures, proving Hayashi's conjecture in specific cases, which relates cycle lengths of permutations to the structure's properties.

Contribution

The paper proves Hayashi's conjecture for certain classes of finite connected quandles, advancing understanding of their cycle structures.

Findings

Hayashi's conjecture holds in particular cases

Cycle type is independent of element choice in finite connected racks

Provides partial validation of a structural property of quandles

Abstract

A rack is a set with a binary operation such that left multiplications are automorphisms of the set and a quandle is a rack satisfying a certain condition. For a finite connected rack the cycle type of the permutation defined by left multiplication by an element is independent from the chosen element. This cycle type is called the profile of the rack. Hayashi conjectured, in the profile of a finite connected quandle, the length of a cycle must divide the length of the largest cycle. In this paper, we prove Hayashi's Conjecture in some particular cases.