Abstract:It has been proven by F. Leong and the first author (J. Algebra {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 $p<q_1<q_2<\cdot \cdot \cdot <q_n$, and \roster \item "(i)" $\alpha \leq 3$, $\beta _i\leq 2$; or \item "(ii)" $p\geq 5$, $\alpha \leq 4$, $\beta _i\leq 2$. \endroster \par 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 {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 $p_1<p_2<\cdot \cdot \cdot <p_n<q$, $q\not \equiv 1(\text {\rm mod} p_i)$, $p_i\not \equiv 1(\text {\rm mod} p_j)$ and the nucleus is not trivial.
Keywords: Moufang loop, order, nonassociative
AMS Subject Classification: Primary 20N05