Homogeneous quandles arising from automorphisms of symmetric groups

TL;DR

This paper explores generalized Alexander quandles derived from symmetric groups, establishing a correspondence between these quandles and conjugacy classes for most symmetric groups between 3 and 30, excluding some specific cases.

Contribution

It introduces a detailed study of generalized Alexander quandles from symmetric groups and proves a one-to-one correspondence with conjugacy classes for a range of n.

Findings

One-to-one correspondence between quandles and conjugacy classes for most symmetric groups 3 ≤ n ≤ 30.

Development of quandle invariants for generalized Alexander quandles.

Discussion of special cases n=6 and n=15.

Abstract

Quandle is an algebraic system with one binary operation, but it is quite different from a group. Quandle has its origin in the knot theory and good relationships with the theory of symmetric spaces, so it is well-studied from points of view of both areas. In the present paper, we investigate a special kind of quandles, called generalized Alexander quandles $Q(G,\psi)$, which is defined by a group $G$ together with its group automorphism $\psi$. We develop the quandle invariants for generalized Alexander quandles. As a result, we prove that there is a one-to-one correspondence between generalized Alexander quandles arising from symmetric groups $\Sf_n$ and the conjugacy classes of $\Sf_n$ for $3 \leq n \leq 30$ with $n \neq 6,15$, and the case $n=6$ is also discussed.