Associativity Of Moufang Loops Of Odd Order
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Date
2017-05
Authors
Lois, Adewoye Ademola
Journal Title
Journal ISSN
Volume Title
Publisher
Universiti Sains Malaysia
Abstract
A loop hL; i is called a Moufang loop if it satisfies the (Moufang) identity (x y)
(z x) = (x (y z)) x. A given Moufang loop may not be a group since it may contain
the elements x;y; z that do not satisfy the (associative) identity (x y) z = x (y z).
However, Moufang’s theorem states that, "if there exist three (fixed) elements x;y; z in
a Moufang loop that associate in some particular order, then hx;y; zi is an associative
subloop." The question on "the associativity of every Moufang loop of order m, for
a particular m 2 Z+" is still open for research. Classification of even order Moufang
loops has more or less been completed, mainly by the efforts of O. Chein and A. Rajah.
So the current focus is on Moufang loops of odd order, particularly constructing classes
of minimally nonassociative Moufang loops, i.e., nonassociative Moufang loops whose
every proper subloop and proper quotient loops are associative. It has been proven by
R. H. Bruck and F. Leong respectively that all Moufang loops of order 33 and p4 (where
p is a prime greater that 3) are associative. However, G. Bol and C. R. B. Wright respectively
also proved the existence of nonassociative Moufang loops of order 34 and
p5 (for any prime p > 3). Also proven by A. Rajah is the existence of nonassociative
Moufang loops of odd order pq3 (p;q are primes) with the (necessary and sufficient)
condition q 1(mod p). Therefore to construct a new class of minimally nonassociative
Moufang loops of odd order m, a necessary condition would be that pq3 divides m
where p and q are distinct odd primes with q 6 1(mod p). W. L. Chee and A. Rajah,
working on this, proved the non-existence of nonassociative Moufang loops of orders
p21
p2n
q3 (p1; p2; pn;q are distinct odd primes with q 6 1(mod pi)), p3q3 (p < q)
and pq4 (p 6= q) respectively. (For the latter two cases, p and q are odd primes with
q 6 1(mod p).)
In this research, the work of the last two authors in precluding the existence of a nonassociative
Moufang loop is expanded to include the following orders (in each case p, q,
p1; ; pn, q1; ;qn are distinct odd primes):
1. p31
p32
p3n
, with pi 6 1(mod pj).
2. p4q3, with 3 < p < q and q 6 1(mod p).
3. p4q31
q3n
, with 3 < p < qi, qi 6 1(mod p) and qi 6 1(mod qj).
4. p21
p22
p2n
1p4n
, with n 2 Z+, 3 p1 < p2 < < pn and pj 6 1(mod pi).
Finally, necessary conditions for the existence of a minimally nonassociative Moufang
loop of order mq4, where q is an odd prime and m is a product of odd primes smaller
than q, are given.
Description
Keywords
Associativity of moufang loops , of odd order