UNIQUELY 2-DIVISIBLE BOL LOOPS

A Class of Loops Categorically Isomorphic to Uniquely 2-divisible Bruck Loops

We define a new variety of loops we call Γ-loops. After showing Γ-loops are power associative, our main goal will be showing a categorical isomorphism between uniquely 2-divisible Bruck loops and uniquely 2-divisible Γ-loops. Once this has been established, we can use the well known structure of Bruck loops of odd order to derive the Odd Order, Lagrange and Cauchy Theorems for Γ-loops of odd or...

Simple Bol Loops

Infinite Simple Bol Loops

If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.

A Class of Finite Simple Bol Loops of Exponent 2

In this paper we give an infinite class of finite simple right Bol loops of exponent 2. The right multiplication group of these loops is an extension of an elementary Abelian 2-group by S5. The construction uses the description of the structure of such loops given by M. Aschbacher (2005). These results answer some questions of M. Aschbacher.

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 order pq. When q does not divide p−1, the only right Bol loop of order pq is the cyclic group of o...