Finite Zassenhaus Moufang Sets With Root Groups of Even Order

Abstract

In [18] 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. τ ∈ Sym(X) which interchanges 0 and ∞ and which maps U ∞ onto U 0 . The Moufang set is also denoted by M (U, τ ), see Section 2.The finite Moufang sets were classified a long time ago using different language. This was done by Hering, Kantor and Seitz [14] (see also the references therein). Their classification uses difficult and long papers such as [19] and [9]. It seems to us that the concept of a Moufang set is the appropiate language to carry out the determination of these groups.De Medts and Segev ([4] and [16]) gave a new proof using this language under the further condition that the Moufang set is special -for the definition of special see the next section. The goal of this paper is to extend their proof to the finite Zassenhaus Moufang sets and thereby giving a partial answer to Question 3 posed by Segev in [16]. A Moufang set is Zassenhaus if G † is a Zassenhaus group, i.e. if in G † there is a non-identy element which fixes two elements in X, but only the identity fixes three elements.The finite Zassenhaus Moufang sets had been determined by Feit [7], Ito [15], Higman [10] and Suzuki [18] in a long proof. There are two families of examples:The set X is just the projective line P(q), q a prime power, and the little projective group is PSL(2, q) in its natural action on X.MSuz(2 2n+1 ): This Moufang set is the natural domain for the Suzuki group Suz(2 2n+1 ) with n ∈ N, see Definition 5.5.In this paper we give an elementary and short proof of the classification of the finite Zassenhaus Moufang sets with root groups of even order. The latter implies that U contains an involution. We distinguish the two cases according to whether U contains a special involution (see Definition 3.6(b)) or not.Theorem 1 Let M (U, τ ) be a finite Zassenhaus Moufang set such that U is of even order. If there is a special involution in U , then M (U, τ ) = M (q) and G † ∼ = PSL 2 (q) with q = |U | = 2 m for some m in N.Theorem 2 Let M (U, τ ) be a finite Zassenhaus Moufang set such that U is of even order. If there is no special involution in U , then M (U, τ ) = MSuz(q) with q 2 = |U |, q an odd power of 2.As a corollary we obtain Corollary 1.1 Let M (U, τ ) be a finite Zassenhaus Moufang set such that U is of even order. Then one of the following holds:(a) U is abelian, M (U, τ ) = M (q) and G † ∼ = PSL 2 (q) for some even prime power q.(b) U is a Suzuki 2-group, M (U, τ ) = MSuz(q) and G † = Suz(q) with q an odd power of 2.Notice that apart from [16] this paper is one of the first discussing not only special but also non-special Moufang s...

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