More Automorphism Groups of Quandles

TL;DR

This paper investigates the automorphism groups of quandles, proving specific isomorphisms for dihedral quandles and analyzing automorphism groups for all quandles up to order 10, providing extensive computational results.

Contribution

It establishes new isomorphism results for displacement groups of dihedral quandles and verifies automorphism groups for all small quandles up to order 10.

Findings

Displacement group of dihedral quandle is isomorphic to rotation groups of polygons.

Automorphism and inner automorphism groups are equivalent for quandles with trivial columns.

Computed automorphism groups for all quandles up to order 10.

Abstract

We prove that the displacement group of the dihedral quandle with n elements is isomorphic to the group generated by rotations of the n/2-gon when n is even and the n-gon when n is odd. We additionally show that any quandle with at least one trivial column has equivalent displacement and inner automorphism groups. Then, using a known enumeration of quandles which we confirm up to order 10, we verify the automorphism group and the inner automorphism group of all quandles (up to isomorphism) of orders less than or equal to 7, compute these for all 115,431 quandles orders 8, 9, and 10, and extend these results by computing the displacement group of all 115,837 quandles (up to isomorphism) of order less than or equal to 10.