The paper surveys the known results concerning the question: “For what values of n does there exist a nonassociative Moufang loop of order n?” Proofs of the newest results for n odd, and a complete resolution of the case n even are also presented.

Moufang sets are the Moufang buildings of rank one. They were introduced by J. Tits [T1] as a tool for studying algebraic groups of relative rank one. They are essentially equivalent to split BN -pairs of rank one, and as such they have been studied extensively. In some sense they are the basic ‘building blocks’ of all split BN -pairs. In the finite case, it had been a major project to classify...

Generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation are found. In particular, the Lie algebra of the multiplication group of the birepresentation is explicitly given. 2000 MSC: 20N05, 17D10, 20G05 Dedicated to Maks A. Akivis on the occasion of his 85th birthday and 65 years of ...

We introduce coordinates for the nite Moufang hexagons and show with a few applications that this approach makes these objects less mysterious.