Generalized Alexander quandles of finite groups and their characterizations

TL;DR

This paper characterizes generalized Alexander quandles of finite groups by relating them to automorphism conjugacy classes, providing specific characterizations for certain groups and small group orders.

Contribution

It offers a new group-theoretic characterization of generalized Alexander quandles, including classifications for simple, symmetric, abelian, and dihedral groups.

Findings

Correspondence between quandle classes and automorphism conjugacy classes for simple groups

Characterizations of quandles for abelian and dihedral groups

Complete classification of quandles from groups with order up to 15

Abstract

The goal of this paper is to characterization generalized Alexander quandles of finite groups in the language of the underlying groups. Firstly, we prove that if finite groups $G$ are simple, then the quandle isomorphic classes of generalized Alexander quandles of $G$ one-to-one correspond to the conjugacy classes of the automorphism groups of $G$. This correspondence can be also claimed for the case of symmetric groups. Secondly, we give a characterization of generalized Alexander quandles of finite groups $G$ under some assumptions in terms of $G$. As corollaries of this characterization, we obtain several characterizations in some particular groups, e.g., abelian groups and dihedral groups. Finally, we perform a characterization of generalized Alexander quandles arising from groups with their order up to $15$.