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Moufang Loops of Odd order \(p^4q^3\)

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 \(q\equiv 1({\text{ mod }}\ p)\)) have been proved to exist. Consider a Moufang loop L of order \(p^{\alpha }q^{\beta }\) ...