Special Moufang Sets, Their Root Groups and Their Μ-maps

We prove Timmesfeld’s conjecture that special abstract rank one groups are quasisimple. We show that in a special Moufang set the root groups are characterized on the one hand by being regular and normal in the point stabilizer, and on the other hand a normal transitive nilpotent subgroup of the point stabilizer is a root group. We prove that if a root group of a special Moufang set contains an involution, then it is of exponent 2. We also show that the root groups are abelian if and only if the so-called μ-maps are involutions.