Bol loops and Bruck loops of order pq

Abstract Right Bol loops are loops satisfying the identity ( ( z x ) y ) x = z ( ( x y ) x ) , and right Bruck loops are right Bol loops satisfying the identity ( x y ) − 1 = x − 1 y − 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 2 − 1 , the only right Bol loop of order pq is the cyclic group of order pq . When q divides p 2 − 1 , there are precisely ( p − q + 4 ) / 2 right Bol loops of order pq up to isomorphism, including a unique nonassociative right Bruck loop B p , 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 p 2 q or p 3 q . When Q = B p , q , the right multiplication group of Q is isomorphic to the semidirect product of Z p × Z p with Z q . Finally, we offer computational results as to the number of right Bol loops of order pq up to isotopy.