Moufang Loops of Order 2 m , m odd

We first show that every Moufang loop L which contains an abelian associative subloop M of index two and odd order must, in fact, be a group. We then use this to settle the question ”For what value of n = 2m, m odd, must a Moufang loop of order n be associative?” Introduction: This paper is motivated by a question asked by Rajah and Jamal in [19]: If L is a Moufang loop of order 2m with an abelian associative subloop M of order m, must L be a group? Generalizing a result of Leong and Teh [13], which gives an affirmative answer in the case that m = p, p an odd prime, Rajah and Jamal prove that the answer is also affirmative if m = p1...p 2 k, or if M ∼= Cp×Cpn . We will show that the answer is affirmative for any M of odd order. Actually, the question raised above stems from other work done by Fook Leong and his students which investigated the question, ”For what integers, n, must every Moufang loop of order n be associative?” The first result in this direction may be found in [6], where it is shown that every Moufang loop of prime order must be a group. In [3], the author extended this result to show that Moufang loops of order p, p, and pq, where p and q are distinct primes, must be associative. Since there are well known nonassociative Moufang loops of order 2 and 3, it would seem that no extension of the results above is possible. However, in [7], Leong showed that a Moufang loop of order p, with p > 3, must be a group. M. Purtill [16] extended the result to Moufang loops of orders pqr, and pq, (p, q and r distinct primes), although the proof of the latter result has a flaw in the case q < p; see [17]. Then Leong and his students produced a spate of papers, [13], [14], [8], [9], [10], culminating in [11], in which Leong and Rajah show that any Moufang loop of order pαq1 1 ...q αn n , with p < q1 < ... < qn odd primes and with α ≤ 3, αi ≤ 2, is a group, and that the same is true with α = 4, provided