On the construction of quandles of order 3n

TL;DR

This paper introduces methods for constructing quandles of order 3n, analyzing their properties such as connectivity, group structure, involutory nature, and Alexander quandle characteristics, with classification up to isomorphism.

Contribution

It provides new construction techniques for quandles of order 3n and characterizes their properties, advancing understanding of their structure and classification.

Findings

Constructed examples of quandles of order 3n

Conditions for quandles to be connected, group, involutory, and Alexander

Classification of quandles up to isomorphism

Abstract

We present methods of constructing examples of quandles of order 3n, where n is greater or equal to 3. The necessary and sufficient conditions for the constructed examples to be (i) connected (ii) group (conjugate) (iii) involutory and (iv) Alexander quandles are examined and presented. Two particular examples from these methods are presented for illustration purpose and their properties are obtained, and these are used in classifying the constructed examples up to isomorphism.