Automorphism groups of quandles arising from groups

Abstract

Let $G$ be a group and $φ\in \Aut(G)$. Then the set $G$ equipped with the binary operation $a\ast b=\phi (a{b}^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, φ)$ and called the generalised Alexander quandle. When $G$ is additive abelian and $φ= -\id_G$, then $\Alex(G, φ)$ 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 $\phi$ is multiplication by a non-trivial unit of $\mathbb{Z}/p\mathbb{Z}$, then $\Aut\big(\Alex(G, φ)\big)$ acts doubly transitively on $\Alex(G, φ)$. This generalises a recent result of \cite{Ferman} for quandles of prime order.

11 pp, minor typo corrected

References in corpus (2)

• Quandle Colorings of Knots and Applications
• Doubly transitive groups and cyclic quandles

Cited by in corpus (6)

• Quandle rings
• Quandle cohomology, extensions and automorphisms
• Free quandles and knot quandles are residually finite
• Orderability of link quandles
• Zero-divisors and idempotents in quandle rings
• Derivations of quandles