Connected quandles and transitive groups

TL;DR

This paper explores the relationship between connected quandles and transitive groups, providing enumeration methods, new proofs for specific orders, and a characterization of affine connected quandles.

Contribution

It establishes a canonical correspondence between connected quandles and transitive group configurations, enabling enumeration and new insights into their structure.

Findings

Efficient enumeration of small order connected quandles.

New proofs for connected quandles of order p and 2p.

A novel characterization of affine connected quandles.

Abstract

We establish a canonical correspondence between connected quandles and certain configurations in transitive groups, called quandle envelopes. This correspondence allows us to efficiently enumerate connected quandles of small orders, and present new proofs concerning connected quandles of order p and 2p. We also present a new characterization of connected quandles that are affine.