Automorphism groups of quandles arising from groups

Abstract

Let G be a group and $\varphi \in {\text {Aut}}(G)$. Then the set G equipped with the binary operation $a*b=\varphi (ab^{-1})b$ gives a quandle structure on G, denoted by ${\text {Alex}}(G, \varphi )$, and called the generalised Alexander quandle of G with respect to $\varphi$. When G is an additive abelian group and $\varphi = -\mathrm {id}_G$, then ${\text {Alex}}(G, \varphi )$ is the well-known Takasaki quandle of G. 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 ${\text {Aut}}\big ({\text {Alex}}(G, \varphi )\big )$ acts doubly transitively on ${\text {Alex}}(G, \varphi )$. This generalises a recent result of Ferman et al. (J Knot Theory Ramifications 20:463–468, 2011) for quandles of prime order.