Associativity Of Moufang Loops Of Odd Order
| dc.contributor.author | Lois, Adewoye Ademola | |
| dc.date.accessioned | 2018-02-08T06:51:30Z | |
| dc.date.available | 2018-02-08T06:51:30Z | |
| dc.date.issued | 2017-05 | |
| dc.description.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. | en_US |
| dc.identifier.uri | http://hdl.handle.net/123456789/5503 | |
| dc.language.iso | en | en_US |
| dc.publisher | Universiti Sains Malaysia | en_US |
| dc.subject | Associativity of moufang loops | en_US |
| dc.subject | of odd order | en_US |
| dc.title | Associativity Of Moufang Loops Of Odd Order | en_US |
| dc.type | Thesis | en_US |
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