Geodesic
Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 301-307.

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It has been proven by F. Leong and the first author (J. Algebra {\bf 190} (1997), 474--486) that all Moufang loops of order $p^\alpha q_1^{\beta_1}q_2^{\beta_2}\cdot \cdot \cdot q_n^{\beta_n}$ where $p$ and $q_i$ are odd primes, are associative if $p$, and \roster \item"(i)" $\alpha\leq 3$, $\beta_i\leq 2$; or \item"(ii)" $p\geq 5$, $\alpha\leq 4$, $\beta_i\leq2$. \endroster The first author also proved that if $p$ and $q$ are distinct odd primes, then all Moufang loops of order $pq^3$ are associative if and only if $q\not\equiv 1(\text{\rm mod}\, p)$ (J. Algebra {\bf 235} (2001), 66--93). In this paper, we prove that all Moufang loops of order $p_1p_2\cdot \cdot \cdot p_nq^3$ where $p_i$ and $q$ are odd primes, are associative if $p_1$, $q\not\equiv 1(\text{\rm mod}\, p_i)$, $p_i\not\equiv 1(\text{\rm mod}\, p_j)$ and the nucleus is not trivial.
Classification : 20N05
Keywords: Moufang loop; order; nonassociative 
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     title = {Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus},
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Rajah, Andrew; Chong, Kam-Yoon. Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 301-307. http://geodesic.mathdoc.fr/item/CMUC_2008__49_2_a11/