Moufang loops of odd order $p_1{p}_2\cdots p_n{q}^3$ with non-trivial nucleus

Rajah, Andrew, and Chong, Kam-Yoon. "Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus." Commentationes Mathematicae Universitatis Carolinae 49.2 (2008): 301-307. <http://eudml.org/doc/250445>.

@article{Rajah2008,
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 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 $q\lnot \equiv 1(\text\{\rm mod\}\, p)$ (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 $p_1<p_2<\cdot \cdot \cdot <p_n<q$, $q\lnot \equiv 1(\text\{\rm mod\}\, p_i)$, $p_i\lnot \equiv 1(\text\{\rm mod\}\, p_j)$ and the nucleus is not trivial.},
author = {Rajah, Andrew, Chong, Kam-Yoon},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Moufang loop; order; nonassociative; nonassociative Moufang loops; orders of finite loops},
language = {eng},
number = {2},
pages = {301-307},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus},
url = {http://eudml.org/doc/250445},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Rajah, Andrew
AU - Chong, Kam-Yoon
TI - Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 2
SP - 301
EP - 307
AB - 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 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 $q\lnot \equiv 1(\text{\rm mod}\, p)$ (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 $p_1<p_2<\cdot \cdot \cdot <p_n<q$, $q\lnot \equiv 1(\text{\rm mod}\, p_i)$, $p_i\lnot \equiv 1(\text{\rm mod}\, p_j)$ and the nucleus is not trivial.
LA - eng
KW - Moufang loop; order; nonassociative; nonassociative Moufang loops; orders of finite loops
UR - http://eudml.org/doc/250445
ER -