Automorphism groups of Quandles

TL;DR

This paper characterizes the automorphism groups of dihedral quandles and provides computational methods for classifying small quandles, advancing understanding of their symmetries and classifications.

Contribution

It proves the automorphism group of dihedral quandles is isomorphic to the affine group and offers algorithms for classifying quandles up to order nine.

Findings

Automorphism group of dihedral quandle is isomorphic to the affine group.

Computed automorphism and inner automorphism groups for all quandles of order six.

Developed an algorithm to classify all quandles up to order nine.

Abstract

We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [9], automorphism groups of quandles (up to isomorphisms) of order less than or equal to 5 were given. With the help of the software Maple, we compute the inner and automorphism groups of all seventy three quandles of order six listed in the appendix of [4]. Since computations of automorphisms of quandles relates to the problem of classification of quandles, we also describe an algorithm implemented in C for computing all quandles (up to isomorphism) of order less than or equal to nine.