Embedding Alexander quandles into groups

For any twisted conjugate quandle Q, and in particular any Alexander quandle, there exists a group G such that Q is embedded into the conjugation quandle of G. 1. EMBEDDABLE QUANDLES A non-empty set Q equipped with a binary operation Q×Q → Q, (x,y) 7→ x∗ y is called a quandle if it satisfies the following three axioms: (1) x∗ x = x (x ∈ Q), (2) (x∗ y)∗ z = (x∗ z)∗ (y∗ z) (x,y,z ∈ Q), (3) For all x ∈ Q, the map Sx : Q → Q defined by y 7→ y∗ x is bijective. Quandles were introduced independently by Joyce [7] and Matveev [9]. Since then, quandles have been important objects in the study of knots and links, set-theoretical solutions of the Yang-Baxter equation, Hopf algebras and many others. We refer to Nosaka [10] for further details of quandles. A map f : Q → Q′ of quandles is called a quandle homomorphism if it satisfies f (x ∗ y) = f (x) ∗ f (y) (x,y ∈ Q). Given a group G, the set G equipped with a quandle operation h ∗ g ≔ g−1hg is called the conjugation quandle of G and is denoted by Conj(G). A quandle Q is called embeddable if there exists a group G and an injective quandle homomorphism Q → Conj(G). Not all quandles are embeddable (see the bottom of §2). In their paper [2], Bardakov-Dey-Singh proposed the question “For which quandles X does there exists a group G such that X embeds in the conjugation quandle Conj(G)?” [2, Question 3.1], and proved that Alexander quandles associated with fixed-point free involutions are embeddable [2, Proposition 3.2]. The following is a list of embeddable quandles of which the author is aware: (1) free quandles and free n-quandles (Joyce [7, Theorem 4.1 and Corollary 10.3]), (2) commutative quandles, latin quandles and simple quandles (Bardakov-Nasybullov [3, §5]), (3) core quandles (Bergman [4, (6.5)]), and (4) generalized Alexander quandles associated with fixed-point free automorphisms (Dhanwani-Raundal-Singh [5, Proposition 3.12]). In this short note, we will show that twisted conjugation quandles, which include all Alexander quandles, are embeddable, thereby generalize the aforementioned result of Bardakov-Dey-Singh. 2020 Mathematics Subject Classification. Primary 20N02; Secondary 08A05, 57K10.