Automorphism groups of quandles arising from groups
Valeriy G. Bardakov, Pinka Dey, Mahender Singh

TL;DR
This paper characterizes automorphism groups of certain algebraic structures called quandles derived from groups, focusing on Takasaki and Alexander quandles, and explores their symmetry properties and automorphism actions.
Contribution
It determines automorphism and inner automorphism groups of Takasaki and Alexander quandles for specific classes of abelian groups, extending previous results on quandles of prime order.
Findings
Automorphism groups of Takasaki quandles of abelian groups with no 2-torsion are determined.
Automorphism groups of Alexander quandles of finite abelian groups with fixed-point free automorphisms are characterized.
Automorphism groups act doubly transitively on the quandles in certain cases, generalizing recent prime order results.
Abstract
Let be a group and . Then the set equipped with the binary operation gives a quandle structure on , denoted by and called the generalised Alexander quandle. When is additive abelian and , then 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 and is multiplication by a non-trivial unit of , then acts doubly transitively on . This generalises a recent result of \cite{Ferman} for quandles of prime…
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