J. Group Theory 15 (2012), 137–149 DOI 10.1515/JGT.2011.100 Journal of Group Theory ( de Gruyter 2012 Moufang loops that are almost groups Wing Loon Chee and Andrew Rajah Abstract. It is known that Moufang loops are closely related to groups as they have many properties in common. For instance, they both have an inverse for each element and satisfy Lagrange’s theorem. In this paper, we study the properties of a class of Moufang loops which are not groups, but all their proper subloops and proper quotient loops are groups. 1 Introduction A group can be alternatively defined as an associative quasigroup—a binary system equipped with divisibility. If a quasigroup further contains an identity element, then it is called a loop. One of the most intensively studied varieties of loops is Moufang loops—loops that satisfy the Moufang identity xy  zx ¼ðx  yzÞx. Moufang loops are closer to groups than general loops and they share many com- mon properties, e.g., the inverse property [2]. In 2005, Grishkov and Zavarnitsine [8] showed that Moufang loops satisfy Lagrange’s theorem, a long-speculated conjecture in Moufang loop theory. Another important theorem is Moufang’s theorem [14]: (i) Any associative triplets generate a group, (ii) Any two elements generate a group (diassociativity). However, Moufang loops are not groups in general. The smallest Moufang loop which is not a group, was constructed in 1971 by Chein and Pflugfelder [5]: a non- associative Moufang loop of order 12. Besides that, classes of non-associative Moufang loops of the following orders were discovered: (i) 2m provided there exists a non-abelian group of order m [3], (ii) 3 4 [1], (iii) p 5 where p is a prime greater than 3 [18], and (iv) pq 3 where p and q are odd primes satisfying q 1 1 ðmod pÞ [16]. Simultaneously, moving in the opposite direction (that is, by proving that all Moufang loops of certain fixed orders are groups), Chein [3] showed that all Moufang loops of orders p, p 2 , p 3 and pq ( p, q primes) are groups. Following that, various authors (Purtill, Leong, Rajah, Chee) have studied Moufang loops of higher orders. The results below summarize their works.