Title

Bol Loops and Bruck Loops of Order pq up to Isotopism

Document Type

Article

Publication Date

3-20-2018

Keywords

Bol loop, Bruck loop, Quadratic field extension, Enumeration, Isotopism

Organizational Units

College of Natual Science and Mathematics, Mathematics

Abstract

Let p>q" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">p>q be odd primes. We classify Bol loops and Bruck loops of order pq up to isotopism. When q does not divide p2−1" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">p2−1, the only Bol loop (and hence the only Bruck loop) of order pq is the cyclic group of order pq. When q divides p2−1" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">p2−1, there are precisely ⌊(p−1+4q)(2q)−1⌋" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">⌊(p−1+4q)(2q)−1⌋ Bol loops of order pq up to isotopism, including a unique nonassociative Bruck loop of order pq.

Publication Statement

Copyright held by author or publisher. User is responsible for all copyright compliance.

This document is currently not available here.

Share

COinS