Automorphism groups of quandles and related groups

Abstract

In this paper we study various questions concerning automorphisms of quandles. For a conjugation quandle 𝑄=Conj(𝐺) of a group G we determine several subgroups of Aut(𝑄) and find necessary and sufficient conditions for these subgroups to coincide with the whole group Aut(𝑄). In particular, we prove that Aut(Conj(𝐺))=Z(𝐺)⋊Aut(𝐺) if and only if either Z(𝐺)=1 or G is one of the groups ℤ2, ℤ22 or ℤ3. 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 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 k-transitive action of Aut(𝑄) for 𝑘≥3.