Moufang Loops Of Odd Order pqr3

Our aim was to study Moufang loops of order pqr³ where p, q and r are distinct odd primes with r ≠ 1 (mod p) and r ≠ 1 (mod q). Since the cases r < p < q and r < q < p had already been resolved previously, we only wished to study the cases r > p or r > q . During the course of research on this \ud problem, having solved it for the particular case of Moufang loops with non-trivial nucleus, we extended the result to the more general case without the condition of having a non-trivial nucleus. Finally, we extended even this result by proving that all Moufang loops of order p₁p₂• • • pᵑq³ where p₁p₂• • • pᵑq³ are distinct odd primes are associative if and only if q ≠ 1(mod pᵢ) for each ᵢ ϵ {1,2,...,n} .\u