FINITE ZASSENHAUS MOUFANG SETS WITH ROOT GROUPS OF EVEN ORDER

In [On a class of doubly transitive groups, Ann. of Math.75 (1962) 104–145] Suzuki classified all Zassenhaus groups of finite odd degree. He showed that such a group is either isomorphic to a Suzuki group or to PSL (2, q) with q a power of 2. In this paper we give another proof of this result using the language of Moufang sets. More precisely, we show that every Zassenhaus Moufang set having root groups of finite even order is either special and thus isomorphic to the projective line over a finite field of even order or is isomorphic to a Suzuki Moufang set. In the final section we discuss generalized Suzuki Moufang sets and see that some properties such as that all involutions in a root group are conjugate hold only in the "ordinary" Suzuki Moufang sets.