Quandles associated with group actions

A quandle is an algebraic system that can be regarded as a generalization of the conjugation operation in groups. In this paper, we establish a Cayley-type embedding theorem for finite quandles. To this end, we study a quandle construction associated with group actions and determine its fundamental structural properties. Applying the construction to the natural action of the symmetric group, we obtain a quandle into which every quandle of cardinality $n$ embeds.