Classification of simple quandles of small order

TL;DR

This work investigates the classification of simple quandles through permutation-group theory by introducing and exploiting quasiprimitive and primitive/quasiprimitive frameworks. It builds a constructive bridge between primitive/quasiprimitive envelopes $(G,\rho)$ and primitive/quasiprimitive quandles, enabling explicit constructions and isomorphism criteria via $\operatorname{Conj}(G,\rho^G)$. Using GAP-based enumeration across groups up to order $4096$, the authors enumerate $268$ primitive quandles in total (with $240$ non-isomorphic) and, in the quasiprimitive setting, construct $991$ quasi-imprimitive quandles (of which $802$ are non-isomorphic), all of which are simple. The results substantially expand the catalog of finite simple and non-affine quandles and tie their structure to the classification of primitive and quasiprimitive groups, providing extensive datasets and computational tools for knot-theoretic invariant studies.

Abstract

In this article, we define quasiprimitive quandles and describe them with the help of quasiprimitive permutation groups. As a consequence, we enumerate finite non-affine simple quandles up to order $4096$.

Theorems & Definitions (26)

• Example 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7
• Theorem 2.8
• Theorem 2.9
• Theorem 2.10