Chee , Wing Loon
(2010)
Classification Of Moufang Loops
Of Odd Order.
PhD thesis, Universiti Sains Malaysia.
Abstract
The Moufang identity (x · y) · (z · x) = [x · (y · z)] · x was first introduced by
Ruth Moufang in 1935. Now, a loop that satisfies the Moufang identity is called
a Moufang loop. Our interest is to study the question: “For a positive integer n,
must every Moufang loop of order n be associative?”. If not, can we construct a
nonassociative Moufang loop of order n?
These questions have been studied by handling Moufang loops of even and
odd order separately. For even order, Chein (1974) constructed a class of nonassociative
Moufang loop, M(G, 2) of order 2m where G is a nonabelian group of
order m. Following that, Chein and Rajah (2000) have proved that all Moufang
loops of order 2m are associative if and only if all groups of order m are abelian.
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