Pseudo-automorphisms and Moufang Loops

An extensive study of Moufang loops is given in [2].1 One defect of that study is that it assumes Moufang's associativity theorem [6], the only published proof of which involves a complicated induction. Using pseudo-automorphisms along with recent methods of Kleinfeld and the author [S], we shall give simple noninductive proofs of three associativity theorems, one of which (Theorem 5.1) generalizes that of Moufang. As shown in [3], still simpler proofs of Moufang's theorem are possible in the commutative case. And, indeed, the following corollary of Theorem 5.3 can be obtained directly from Lemmas 2.1, 2.2: Every associative subset of a commutative Moufang loop G is contained in an associative subloop of G. The present methods represent a considerable improvement over those of [2] (in particular, pseudo-automorphisms have displaced the cumbersome autolopisms) and the paper should serve as an introduction to the theory of Moufang loops. There is little overlapping, except possibly in §2, but we have added (Theorem 4.1) a more aesthetic proof of the fact that the nucleus (previously called the associator) of a Moufang loop is a normal subloop.