Bol loops and Bruck loops of order $pq$

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 $p^{2}q$ or $p^{3}q$. 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.

23 pages; to appear in J. Algebra

References in corpus (1)

• Three lectures on automorphic loops

Cited by in corpus (1)

• Half-isomorphisms whose inverses are also half-isomorphisms