CLASSIFICATION OF MINIMAL NONASSOCIATIVE MOUFANG LOOPS OF ORDER pq3

Abstract

An open problem in the theory of Moufang loops is to classify those loops which are minimally nonassociative, that is, loops which are nonassociative but where all proper subloops are associative. A related question is to classify all integers n for which a minimally nonassociative loop exists. In [Possible orders of nonassociative Moufang loops, Comment. Math. Univ. Carolin.41(2) (2000) 237–244], O. Chein and the third author showed that a minimal nonassociative Moufang loop of order 2q3can be constructed by using a non-abelian group of order q3. In [Moufang loops of odd order pq3, J. Algebra235 (2001) 66–93], the third author also proved that for odd primes p < q, a nonassociative Moufang loop of order pq3exists if and only if q ≡ 1 ( mod p). Here we complete the classification of minimally nonassociative Moufang loops of order pq3for primes p < q.