Generalized Alexander quandles of finite groups and their characterizations (Q6595559)

A quandle is a set \(Q\) equipped with a binary operation \( \ast \) where \( \ast \) is idempotent, the equation \(x\ast b=a\) has a unique solution for \(x\), for all \(a,b\in Q\) and \((x\ast y)\ast z=(x\ast z)\ast(y\ast z)\). \N\NLet \(G\) be a group, \(K\) a subgroup of \(G\) and \(\psi\) an automorphism group of \(G\). The triplet \((G,K,\psi)\) is called a quandle triplet if \(K\subset \mathrm{Fix}(\psi,G)=\{x\in G:\psi(x)=x\}\). When \(K=\{e\}\) is the trivial subgroup of \(G\), \(Q(G,\psi)\) is called the generalized Alexander quandle.\N\NThe paper studies these groups and provides several characterizations of them. Particularly, up to order \(15\), the generalized Alexander quandles are characterized.\N\NSome useful references are [\textit{D. Joyce}, J. Pure Appl. Algebra 23, 37--65 (1982; Zbl 0474.57003); \textit{Y. Ishihara} and \textit{H. Tamaru}, Proc. Am. Math. Soc. 144, No. 11, 4959--4971 (2016; Zbl 1406.53058); \textit{A. Higashitani} and \textit{H. Kurihara}, Commun. Algebra 51, No. 4, 1413--1430 (2023; Zbl 1512.20223); \textit{M. Bonatto}, Monatsh. Math. 191, No. 4, 691--717 (2020; Zbl 1481.20218)].