Bol loops and Bruck loops of order $pq$

TL;DR

This paper classifies right Bol loops of order pq for odd primes p and q, revealing their structure, automorphism groups, and providing computational enumeration results, advancing the understanding of these algebraic loops.

Contribution

It provides a complete classification of right Bol loops of order pq, including the unique nonassociative case, and analyzes their nuclei and multiplication groups.

Findings

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

When q divides p^2-1, there are (p-q+4)/2 right Bol loops of order pq.

The right multiplication group of a nonassociative right Bol loop of order pq is isomorphic to a semidirect product of Z_p×Z_p with Z_q.

Abstract

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 $(xy)^{-1} = x^{-1}y^{-1}$. 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 $p^2q$ or $p^3q$. When $Q=B_{p,q}$, the right multiplication group of $Q$ is isomorphic to the semidirect product of $\mathbb{Z}_p\times \mathbb{Z}_p$ with $\mathbb{Z}_q$. Finally, we offer computational results as to the number of right Bol loops of order $pq$ up to isotopy.