Classification of Finite Alexander Quandles

TL;DR

This paper provides a classification method for finite Alexander quandles by analyzing their module structures, leading to explicit criteria for isomorphism, duality, and connectivity, and enumerates such quandles up to fifteen elements.

Contribution

It introduces a module-theoretic approach to classify finite Alexander quandles and offers a practical procedure for their enumeration and analysis.

Findings

Isomorphism criteria for finite Alexander quandles based on module isomorphisms

Explicit conditions for when linear quandles are isomorphic or dual

Enumeration of distinct and connected Alexander quandles up to 15 elements

Abstract

Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander quandles of the form Z_n[t,t^-1]/(t-a) where gcd(n,a)=1 (called linear quandles) are isomorphic, as well as specific conditions on when two linear quandles are dual and which linear quandles are connected. We apply this result to obtain a procedure for classifying Alexander quandles of any finite order and as an application we list the numbers of distinct and connected Alexander quandles with up to fifteen elements.