Automorphism groups of quandles and related groups

Abstract

In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q=\mathrm{Conj}(G)$ of a group $G$ we determine several subgroups of $\mathrm{Aut}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group $\mathrm{Aut}(Q)$. In particular, we prove that $\mathrm{Aut}(\mathrm{Conj}(G))=\mathrm{Z}(G)⋊\mathrm{Aut}(G)$ if and only if either $\mathrm{Z}(G)=1$ or $G$ is one of the groups ${\mathbb{Z}}_2$, ${\mathbb{Z}}_2^2$ or ${\mathbb{Z}}_3$. For a big list of Takasaki quandles $T(G)$ of an abelian group $G$ with $2$-torsion we prove that the group of inner automorphisms $\mathrm{Inn}(T(G))$ is a Coxeter group. We study automorphisms of certain extensions of quandles and determine some interesting subgroups of the automorphism groups of these quandles. Also we classify finite quandles $Q$ with $3\le k$-transitive action of $\mathrm{Aut}(Q)$.

19 pages