Automorphism groups of quandles arising from groups

TL;DR

This paper characterizes automorphism groups of certain algebraic structures called quandles derived from groups, focusing on Takasaki and Alexander quandles, and explores their symmetry properties and automorphism actions.

Contribution

It determines automorphism and inner automorphism groups of Takasaki and Alexander quandles for specific classes of abelian groups, extending previous results on quandles of prime order.

Findings

Automorphism groups of Takasaki quandles of abelian groups with no 2-torsion are determined.

Automorphism groups of Alexander quandles of finite abelian groups with fixed-point free automorphisms are characterized.

Automorphism groups act doubly transitively on the quandles in certain cases, generalizing recent prime order results.

Abstract

Let $G$ be a group and $\varphi \in \Aut(G)$. Then the set $G$ equipped with the binary operation $a*b=\varphi(ab^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, \varphi)$ and called the generalised Alexander quandle. When $G$ is additive abelian and $\varphi = -\id_G$, then $\Alex(G, \varphi)$ is the well-known Takasaki quandle. In this paper, we determine the group of automorphisms and inner automorphisms of Takasaki quandles of abelian groups with no 2-torsion, and Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. As an application, we prove that if $G\cong (\mathbb{Z}/p \mathbb{Z})^n$ and $\varphi$ is multiplication by a non-trivial unit of $\mathbb{Z}/p \mathbb{Z}$, then $\Aut\big(\Alex(G, \varphi)\big)$ acts doubly transitively on $\Alex(G, \varphi)$. This generalises a recent result of \cite{Ferman} for quandles of prime order.