Bol loops and Bruck loops of order $pq$ up to isotopism

TL;DR

This paper classifies Bol and Bruck loops of order pq for odd primes p and q, revealing conditions under which they are cyclic or nonassociative, and enumerating their isotopism classes.

Contribution

It provides a complete classification of Bol and Bruck loops of order pq up to isotopism, including enumeration and identification of unique nonassociative cases.

Findings

When q does not divide p^2-1, the only Bol and Bruck loop is cyclic of order pq.

If q divides p^2-1, there are finitely many Bol loops up to isotopism.

There exists a unique nonassociative Bruck loop of order pq.

Abstract

Let $p>q$ be odd primes. We classify Bol loops and Bruck loops of order $pq$ up to isotopism. When $q$ does not divide $p^2-1$, the only Bol loop (and hence the only Bruck loop) of order $pq$ is the cyclic group of order $pq$. When $q$ divides $p^2-1$, there are precisely $\lfloor(p-1+4q)(2q)^{-1}\rfloor$ Bol loops of order $pq$ up to isotopism, including a unique nonassociative Bruck loop of order $pq$.