Paramedial quasigroups of prime and prime square order

TL;DR

This paper classifies paramedial quasigroups of prime and prime square order, providing counts, isomorphism classifications, and identifying simple cases, thereby advancing the understanding of their algebraic structure.

Contribution

It offers a complete enumeration and classification of paramedial quasigroups of prime and prime square order, including simple cases, up to isomorphism.

Findings

There are 2p-1 paramedial quasigroups of order p for each odd prime p.

There are 6p^2 - p - 1 paramedial quasigroups of order p^2.

The paper identifies all simple paramedial quasigroups within these classifications.

Abstract

We prove that, for every odd prime number $p$, there are $2p-1$ paramedial quasigroups of order $p$ and $6p^2-p-1$ paramedial quasigroups of order $p^2$, up to isomorphism. We present a complete list of those which are simple.