A class of simple Moufang loops

Abstract

As we shall see in the course of our proof, the present theorem is a nonassociative analogue of the well known results on the special projective group PSL(n, K) (see [4, p. 44]). In ?5, we shall prove that the Cayley-Dickson numbers of norm 1 over the real field R* (modulo their center) are simple and indicate how this is the best possible result. Our results will yield finite, not-associative, simple Moufang loops whose possible orders are (21n-22n) and 2-1(p7n_ p3n) if p is an odd prime. Thus we obtain a simple, not-associative, Moufang loop of order 120. Although we have tried to make this paper reasonably self contained, some of the results by Bruck (2) on Moufang loops will be used without reference.