ABSTRACT. The main result of this paper is the determination of all non- 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.Key words and phrases. Moufang loop, diassociative, subloop, order, Lagrange's theorem, Sylow's theorem, center, nucleus.(1) In order to know the Moufang loops of order n, one must know the groups of order £ n/2. Since the groups of order s 15 are well known, whereas the groups of order 16 and larger start getting messy, 31 was chosen as our stopping point. However, we hope to extend our work in a future paper.
