Constructions and automorphisms of quandles

Abstract

We prove that if M is an abelian group and X = Alex(M, phi) is a Latin Alexander quandle with id - phi in Aut(M), then Aut(X) is isomorphic to M semidirect C_Aut(M)(phi), Inn(X) is isomorphic to M semidirect <phi> and Out(X) is isomorphic to C_Aut(M)(phi)/<phi>. As a consequence, we obtain a description of the automorphism groups of direct products of the cyclic commutative medial quandles C_{2n+1} and the dihedral quandles R_{2n+1}, and we find their orders. Furthermore, we correct a result of Bardakov-Fedoseev [Algebra and Logic 63 (2024), no. 2, 111-142] on multiplication of quandles and prove that the Borodin-Neshchadim-Simonov and Markhinina-Nasybullov quandles can be constructed from conjugation, core, and generalized Alexander quandles, using the operation of multiplication of quandles. Finally, we consider the quotients of a quandle Q, namely the medialization Q^med, the commutativization Q^com, and the trivialization Q^tr. We prove that the conjugation quandle Conj(G) is medial if and only if gamma_2(G) <= Z(G); we find bounds for |Conj(G)^med| and prove that Core(G)^com is isomorphic to Core(G/<<g^3>>). We describe commutative generalized Alexander quandles and, as a consequence, find the commutativization of the dihedral quandle: R_n^com is isomorphic to R_gcd(n,3).