Classification Of Moufang Loops Of Odd Order - Chee , Wing Loon (2010)

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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.