Proper Moufang Sets with Abelian Root Groups Are Special

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 split BN -pairs of rank one. This project culminated in [HKSe]. Moufang sets are also essentially equivalent to Timmesfeld’s ‘abstract rank one groups’ (see [Ti1]). In recent years there has been a revived interest and significant progress in this area; for additional information, see the bibliography at the end of the paper. Let us recall that a Moufang set is essentially a doubly transitive permutation group such that the point stabilizer contains a normal subgroup which is regular on the remaining points. These regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. In [DW, DS1, DS3] the notation M(U, τ) is used for a Moufang set (and this notation is of course explained there). The group U in this notation is isomorphic to any one of the root groups of the Moufang set. Recall that M(U, τ) is special iff (−a)τ = −(aτ), for all a ∈ U∗. We say that M(U, τ) is a sharply 2-transitive Moufang set if its little projective group is sharply 2-transitive. Recall that a Moufang set is proper if it is not sharply 2-transitive. In this paper we prove: