Moufang loops of odd order p 1p2 ••• p,q
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Date
2007
Authors
Kam Yoon, Chong
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Abstract
A binary system (L,-) in which specification of any two of the elements x, y and z in
the equation x · y = z uniquely determines the third element is called a quasigroup. If
furthermore it contains a (two-sided) identity element, then it is called a loop. A
Moufang loop is a loop which satisfies the Moufang identity:
(X· y) · ( Z· X)= [X· (y · Z) J·X .
Nonassociative Moufang loops of orders 24
, 34 and p 5 (p > 3) are known to exist. In
197 4, 0. Chein proved that all Moufang loops of orders p, p 2
, pq and p 3 are groups
when p and q are primes (see [4]).
It was proven by F. Leong and A. Rajah· (1997) that all Moufang loops of odd order
pa q/'q/2
• • • q/" are associative if p and q; are odd primes with p < q1 < q2 < · · · < qn,
and
(i) a ::;3, fJ; :S2; or
(ii) p ?.. 5, a::; 4, /3; ::; 2 (see [15]).
A. Rajah (2001) proved that if p and q are distinct odd primes, then all the Moufang
loops of order pq3 are groups if and only if q ;/= l(mod p ).
The aim of our research is to study Moufang loops of odd order p1p2 • • • pnq3 where P;
and q are primes, 2 < p1 < p2 < · · · < Pn < q , q =/= l(mod p) and P; =!= l(mod p1 ) for
i, j E {1, 2, · · ·, n} . Before we managed to prove that all such Moufang loops are groups,
we reduced the problem above into a smaller problem so that it is more easily solved.
In Chapter 3, we prove that all Moufang loops of order pqr3
, where p , q and r are
odd primes, p < q < r , q =1= l(mod p) , r =!= l(mod p) and r =/= l(mod q) are associative.
In Chapter 4, we extend the result in Chapter 3 to Moufang loops of odd order
P1P2 • • • pnq3
, where P; and q are primes, 2 < p1 < p2 < ·· · < Pn < q , q =/= l(mod P;)
and P; =!= l(mod p1) for i, j E {1, 2, · · ·, n} , and prove that all such Moufang loops are
associative.
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Keywords
Loops , P 1P2 ••• P,q