In this paper we complete the classification of special Moufang sets M(U , τ ) with abelian Hua subgroup. It is already known that if U has odd characteristic, then M(U , τ ) is the projective line over a commutative field. We present a similar result which also deals with special Moufang sets of even characteristic.

We prove that the unipotent horocyclic group of a Moufang twin tree of prime order is nilpotent of class at most 2.

We clarify the notion of effective equivalence and characterize geometrically the effectively equivalent permutation groups. In particular, we present examples showing that the latter do not correspond to affinely equivalent polytopes thereby answering Question 2.12 of [BHNP09]. We apply our characterization to our examples.

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...

We consider a rank one group G = A, B which acts cubically on a module V , this means [V, A, A, A] = 0 but [V, G, G, G] = 0. We have to distinguish whether the group A 0 := C A ([V, A]) ∩ C A (V /C V (A)) is trivial or not. We show that if A 0 is trivial, G is a rank one group associated to a quadratic Jordan division algebra. If A 0 is not trivial (which is always the case if A is not abelian), then A 0 defines a subgroup G 0 of G which acts quadratically on V . We will call G 0 the quadratic kernel of G. By a result of Timmesfeld we have G 0 ∼ = SL 2 (J, R) for a ring R and a special quadratic Jordan division algebra J ⊆ R. We show that J is either a Jordan algebra contained in a commutative field or a hermitian Jordan algebra. In the second case G is the special unitary group of a pseudo-quadratic form π of Witt index 1, in the first case G is the rank one group for a Freudenthal triple system. These results imply that if (V, G) is a quadratic pair such that no two distinct root groups commute and charV = 2, 3, then G is a unitary group or an exceptional algebraic group. ContentsChapter 1. Introduction Chapter 2. Preliminaries 2.1. Moufang sets 2.2. Rank one groups 2.3. Some ring theory 2.4. Quadratic Jordan algebras 2.5. Envelopes of special quadratic Jordan algebras 2.6. Involutory sets and pseudo-quadratic forms 2.7. Quadrangular algebras and Moufang quadrangles 2.8. Freudenthal triple systems and structurable algebras Chapter 3. Cubic Action Chapter 4. Examples for cubic modules 4.1. Pseudoquadratic spaces 4.2. Adjoint action 4.3. The Tits-Kantor-Koecher module 4.4. Quadratic pairs without commuting root groups 4.5. Quadratic spaces 4.6. Connection to Moufang Quadrangles 4.7. Suzuki and Ree groups Chapter 5. On the structure of a cubic module Chapter 6. Construction of irreducible submodules Chapter 7. Cubic rank one groups with trivial kernel Chapter 8. A characterisation of the adjoint module Chapter 9. Cubic rank one groups with non-trivial kernel Chapter 10. Cubic rank one groups with hermitian quadratic kernel Chapter 11. The construction of a pseudoquadratic space Chapter 12. Cubic rank one groups with commutative quadratic kernel Bibliography v CHAPTER 1