Automorphism groups of quandles arising from groups

Abstract

Let G be a group and 𝜑∈Aut(𝐺). Then the set G equipped with the binary operation 𝑎∗𝑏=𝜑(𝑎𝑏−1)𝑏 gives a quandle structure on G, denoted by Alex(𝐺,𝜑), and called the generalised Alexander quandle of G with respect to 𝜑. When G is an additive abelian group and 𝜑=−id𝐺, then Alex(𝐺,𝜑) 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≅(ℤ/𝑝ℤ)𝑛 and 𝜑 is multiplication by a non-trivial unit of ℤ/𝑝ℤ, then Aut(Alex(𝐺,𝜑)) acts doubly transitively on Alex(𝐺,𝜑). This generalises a recent result of Ferman et al. (J Knot Theory Ramifications 20:463–468, 2011) for quandles of prime order.