Medial quasigroups of prime square order

TL;DR

This paper determines the exact number of medial quasigroups of order p^2 for any prime p, providing a complete classification up to isomorphism.

Contribution

It offers a precise enumeration of medial quasigroups of order p^2, a previously unresolved problem in algebraic structure classification.

Findings

Number of medial quasigroups of order p^2 is 2p^4 - p^3 - p^2 - 3p - 1.

Provides classification up to isomorphism for these quasigroups.

Establishes a formula valid for all primes p.

Abstract

We prove that, for any prime $p$, there are precisely $2p^4-p^3-p^2-3p-1$ medial quasigroups of order $p^2$, up to isomorphism.