BOL LOOPS AND BRUCK LOOPS OF ORDER pq

Right Bol loops are loops satisfying the identity ((zx)y)x = z((xy)x), and right Bruck loops are right Bol loops satisfying the identity (xy)−1 = x−1y−1. Let p and q be odd primes such that p > q. Advancing the research program of Niederreiter and Robinson from 1981, we classify right Bol loops of order pq. When q does not divide p−1, the only right Bol loop of order pq is the cyclic group of order pq. When q divides p − 1, there are precisely (p− q + 4)/2 right Bol loops of order pq up to isomorphism, including a unique nonassociative right Bruck loop Bp,q of order pq. Let Q be a nonassociative right Bol loop of order pq. We prove that the right nucleus of Q is trivial, the left nucleus of Q is normal and is equal to the unique subloop of order p in Q, and the right multiplication group of Q has order pq or pq. When Q = Bp,q, the right multiplication group of Q is isomorphic to the semidirect product of Zp × Zp with Zq. Finally, we offer computational results as to the number of right Bol loops of order pq up to isotopy.