Automorphism groups of quandles arising from groups

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.Comment: 11 pp, minor typo correcte

Quandle rings

“…In [15], a description of the automorphism group of Alexander quandles was determined, and an explicit formula for the order of the automorphism group was given for finite case. In [3], some structural results are obtained for the group of automorphisms and inner automorphisms of generalised Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. This work was extended in [4], wherein several interesting subgroups of automorphism groups of conjugation quandles of groups are determined.…”

Automorphism groups of quandles and related groups

“…As we already mentioned, in [3,Proposition 4.7] it is proved that the group Aut (Conj(G)) contains a subgroup isomorphic to the semidirect product Z(G) ⋊ Aut(G). The following theorem describes all finite groups which satisfy the equality Aut (Conj(G)) = Z(G) ⋊ Aut(G), what gives the answer to [3,Problem 4.8].…”

“…The Takasaki quandle T (G) is a particular case of the Alexander quandle Alex(G, ϕ) for ϕ : x → x −1 . Alexander quandles were studied, for example, in [3,9,10].…”

Free quandles and knot quandles are residually finite

“…Moreover, semi-direct product of residually finite groups is residually finite. By [1,Theorem 4.2], Aut Core(G) ∼ = G⋊Aut(G), and hence Aut Core(G) is residually finite.…”