Homogeneous quandles arising from automorphisms of symmetric groups

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 $n eq 6,15$, and the case $n=6$ is also discussed.