Constructions and automorphisms of quandles

@misc{bardakov2026constructions,
  title = {{Constructions and automorphisms of quandles}},
  author = {Valeriy G. Bardakov and Birama Sangare},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18676},
  eprint = {2610.18676},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18676},
}

TY  - UNPB
TI  - Constructions and automorphisms of quandles
AU  - Valeriy G. Bardakov
AU  - Birama Sangare
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18676
UR  - https://arxiv.org/abs/2610.18676
AB  - 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 and Out(X) is isomorphic to C_Aut(M)(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/<>). 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).
ER  -