A Simplified Proof of Moufang’s Theorem

Moufang’s theorem states that if Q is a Moufang loop with elements x, y and z such that x ·yz = xy · z, then these three elements generate a subgroup of Q. The paper contains a new proof of this theorem that is shorter and more transparent than the standardly used proof of Bruck. A loop is a binary structure with a unit such that the equations ax = b and ya = b have unique solutions x = a\b and y = b/a. The inverses x\1 and 1/x may differ, but if they agree, we denote them by x−1. A loop is called Moufang if it satisfies the three equivalent identities x(y · xz) = (xy · x)z, (zx · y)x = z(x · yx) and xy · zx = x(yz · x). At the very foundations of Moufang loop theory there is a theorem that states that if x, y and z associate, then they generate a subgroup. This statement is called Moufang’s theorem. The standardly used proof of Bruck is a bit of an obstacle to anybody who wishes to learn more about Moufang loops. The problem is not really the length, but rather the technicality, which makes it hard to identify the principles that are behind the proof. The purpose of this note is to show that Moufang’s theorem can be proved straightforwardly with clearly separated ingredients. Facts needed for the proof can be classified as (1) general properties of autotopisms in Moufang loops, (2) relationships between inner mappings that allow a circular shift of an associative triple and (3) a combinatorial observation that in a nonempty word over letters a, b and c, either one of the terminal symbols is in {b, c} or both terminal symbols are equal to a. All these ingredients are present in a varying degree of explicitness in Bruck’s proof. The proof presented here only organizes them in a different way. Technically, Lemma 4 and the proof of Proposition 1 are new. Everything else, the proofs included, is well known and appears just for the sake of completeness. Following a suggestion of the referee, the equivalence of the three Moufang identities is proved at the end of the paper (Proposition 2), as a kind of appendix. Permutations Rx : y → yx and Lx : y → xy of a loop Q are known as the right and left translations of x, respectively. A loop Q is called left alternative if x · xy = xx · y for all x, y ∈ Q. Right alternative loops satisfy yx · x = y · xx, and Received by the editors December 31, 2009 and, in revised form, March 12, 2010. 2010 Mathematics Subject Classification. Primary 20N05; Secondary 08A05.