# Moufang Loops of Odd order $p^4q^3$

Bulletin of the Malaysian Mathematical Sciences Society, Feb 2017

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 $p^{\alpha }q^{\beta }$ where p and q are odd primes with $p<q$, $q\not \equiv 1 ({\text{ mod }}\ p)$ 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 $\beta \le 3$.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs40840-017-0471-2.pdf

Andrew Rajah, Lois Adewoye Ademola. Moufang Loops of Odd order $p^4q^3$, Bulletin of the Malaysian Mathematical Sciences Society, 2017, 1-12, DOI: 10.1007/s40840-017-0471-2