# Principal and Doubly Homogeneous Quandles

## TL;DR

This paper characterizes principal and doubly homogeneous quandles, showing their structure, finiteness, and providing a classification of certain connected cyclic quandles, thus advancing the algebraic understanding of these mathematical objects.

## Contribution

It introduces a comprehensive classification of finite doubly homogeneous quandles and describes the structure of principal and connected quandles.

## Key findings

- Connected quandles decompose into principal quandles.
- Finite simple affine quandles are characterized by double homogeneity.
- Complete classification of finite doubly-homogeneous quandles.

## Abstract

In the paper we describe the class of principal quandles and show that connected quandles can be decomposed as a disjoint union of principal quandles. We also prove that simple affine quandles are finite and they can be characterized among finite simple quandles by several different equivalent properties, as for instance being doubly homogeneous (i.e. having a doubly transitive automorphism group). A complete description of finite doubly-homogeneous quandles is provided extending the result of \cite{V} and solving \cite[Problem 6.7]{2trans}. We also provide a classification of connected cyclic quandles with several fixed points independently from \cite{CQ}.