Moufang Loops of Odd order p^4q^3

Abstract

It is known that all Moufang loops of order p^4 are associative if p is a prime greater than 3. Also, nonassociative Moufang loops of order p^5 (for all primes p) and pq^3 (for distinct odd primes p and q, with the necessary and sufficient condition qequiv 1({text{ mod }} p)) have been proved to exist. Consider a Moufang loop L of order p^{alpha }q^{beta } where p and q are odd primes with p<q, qnot equiv 1 ({text{ mod }} p) and alpha ,beta in {mathbb {Z}}^+. It has been proved that L is associative if alpha le 3 and beta le 3. In this paper, we extend this result to the case p>3, alpha le 4 and beta le 3.