Moufang Loops of Odd Order p1p2... pnq^3

It has been proved that for distinct odd primes p1, p2...pn and q, all Moufang loops of order p1p2...pnq^3 are associative if: (1) q ≠ 1 (mod p1) and for each i 1, q^2 ≠ 1 (mod pi); or (2) p1 p2 ... pn q, q ≠ 1 (mod pi), pi ≠ 1 (mod pj ) for all i; j, and the nucleus is not trivial. In this paper, we extend these results by giving a complete resolution for Moufang loops of odd order p1p2...pnq^3.