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

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 $pq$.