# Bol Loops and Bruck Loops of Order pq up to Isotopism

3-20-2018

Article

Mathematics

## Keywords

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

## 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.

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