Automorphisms and antiautomorphisms of quandles
Birama Sangare
TL;DR
The paper investigates when automorphisms and antiautomorphisms of a group $G$ induce corresponding maps on several group-derived quandles, including $Conj_m(G)$, $Core(G)$, and $Alex(G,\phi)$, as well as various verbal quandle variants. It introduces concrete constructions such as the group $H=\{f_a: G\to G\,|\, a\in Z(G)\}$ and its semidirect product with $\mathrm{Aut}(G)$, and shows $H\rtimes \mathrm{Aut}(G) \le \mathrm{Aut}(\mathrm{Conj}_m(G))$, with an analogous result $G^{op}\rtimes C_{\mathrm{Aut}(G)}(\phi) \le \mathrm{Aut}(\mathrm{Alex}(G,\phi))$, thereby extending known bounds. The authors also prove notable negative results (e.g., no antiautomorphisms for the dihedral quandle $\mathrm{R}_n$ when $n\neq 3$) and construct new automorphisms of $Core(G)$ and $Alex(G,\phi)$ not arising from $G$ via explicit map families, highlighting the richer automorphism structure of quandles. Across sections, they classify when automorphisms or antiautomorphisms originate from or are constrained by centrality and commutator conditions, and they extend these analyses to one-parameter verbal quandles $P_i$ and their analogues $Q_i$, clarifying how centrality and abelianness influence inducibility. Collectively, the results provide a comprehensive framework for understanding the interplay between group automorphisms and the automorphism/antiautomorphism structures of associated quandles with concrete algebraic criteria.
Abstract
In this paper we provide the conditions under which an automorphism or an antiautomorphism of a group $G$ induces an automorphism or an antiautomorphism of the $m$-conjugation quandle $\operatorname{Conj_{m}}(G),\,\, m\in \mathbb{Z} $, the core quandle $\operatorname{Core}(G)$, the generalized Alexander quandle $\operatorname{Alex}(G,φ)$ where $φ\in \operatorname{Aut}(G)$ and some others. We also construct automorphisms of these quandles that do not originate from $G$.