Moufang Loops of Small Order

The main result of this paper is the determination of all nonassociative Moufang loops of orders *31. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange's theorem and Sylow's main theorem. In addition to finding the loops referred to above, we prove that Moufang loops of orders p, p , p or pq (for p and q prime) must be groups. Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2. L Introduction. In studying algebraic objects, it is frequently useful to have many examples at one's fingertips. In the case of Moufang loops that are not groups, the scarcity of manageable examples is one of the difficulties that we have encountered. It is the purpose of this paper to begin to remedy this situation by finding all Moufang loops of order < 31.0) There are 13 such loops-one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28. The order structures, nuclei and subloops of these loops are given (Tables 3, 4 and 5). All of the loops are G-loops (i.e. they are isomorphic to all of their loop isotopes) and they are solvable. Lagrange's theorem and Sylow's main theorem hold in all of them. In terms of the M^-laws of Pflugfelder [10], some of the loops are M}-loops, some are M7-loops, and some are strictly Moufang. In the course of studying these loops, we find a general method of constructing nonassociative Moufang loops as extensions of groups (see Theorem 1). We also prove that, for p and q being primes, Moufang loops of order pq or of order p" for n < 3 are groups. Presented to the Society, July 15, 1971; received by the editors November 15, 1971. AUS (MOS) subject classifications (1970). Primary 20N05.