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 orde...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2017
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| Sorozat: | JOURNAL OF ALGEBRA
473 |
| Tárgyszavak: | |
| doi: | 10.1016/j.jalgebra.2016.11.023 |
| mtmt: | 3181568 |
| Online Access: | http://publicatio.bibl.u-szeged.hu/29125 |
| Tartalmi kivonat: | 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 p2−1, the only right Bol loop of order pq is the cyclic group of order pq. When q divides p2−1, 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 p2q or p3q. 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. © 2016 Elsevier Inc. |
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| Terjedelem/Fizikai jellemzők: | 481-512 |
| ISSN: | 0021-8693 |