Special Moufang sets with abelian Hua subgroup

Grüninger, Matthias

doi:10.1016/j.jalgebra.2009.12.016

Cited by 5 publications

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References 5 publications

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“…For the remaining implication (d) to (a) we use the fact that every Jordan division algebra J defines a special Moufang set M(J) and the Hua group of M(J) is just the inner structure group of J ([12], 4.1 and 4.2). Thus (d) to (a) is a consequence of 6.1 in [12] and the main theorem of [15].…”

Section: Envelopes Of Special Quadratic Jordan Algebrasmentioning

confidence: 85%

On cubic action of a rank one group

Grüninger¹

2011

Preprint

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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

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Section: Envelopes Of Special Quadratic Jordan Algebrasmentioning

confidence: 85%

On cubic action of a rank one group

Grüninger¹

2011

Preprint

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“…In particular, the field √ k consisting of all the elements of an algebraic closure of k whose square belongs to k, is an extension of k of degree 2: it coincides with F q (( √ t)). Therefore, Part (3) of the main theorem from [Grü10] ensures that the little projective group G + is isomorphic to PSL 2 (k) and the Moufang set ∂T is equivariantly isomorphic to the projective line over k. Since the µ-maps are continuous by Proposition 3.8, the isomorphism between G + and PSL 2 (k) must be a homeomorphism. We can now conclude the proof in the same way as in the characteristic = 2 case.…”

Section: 2mentioning

confidence: 99%

Trees, contraction groups, and Moufang sets

Caprace¹,

Medts²

2013

Duke Math. J.

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We study closed subgroups G of the automorphism group of a locally finite tree T acting doubly transitively on the boundary. We show that if the stabiliser of some end is metabelian, then there is a local field k such that PSL2(k)≤G≤PGL2(k). We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if G is (virtually) a rank one simple p-adic analytic group for some prime p. A key point is that if some contraction group is closed, then G is boundary-Moufang, meaning that the boundary ∂T is a Moufang set. We collect basic results on Moufang sets arising at infinity of locally finite trees, and provide a complete classification in case the root groups are torsion-free

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“…Thus M (U, τ ) is special and U is of exponent 2. By 4.6 (c) the Hua subgroup H is abelian, thus the theorem follows with [8] (alternatively, we can apply [4]). In the begining of VIII, 7 in [12] it is shown that in fact we may assume (c') there is a cyclic subgroup of Aut(G) which permutes the involutions transitively.…”

Section: Zassenhaus Moufang Sets With |U| Evenmentioning

confidence: 99%

“…The proof of Theorem 1 uses only the language of Moufang sets. The strategy is to show that U is an elementary abelian 2-group and then to quote [8] or [4].…”

Section: Introductionmentioning

confidence: 99%

Finite Zassenhaus Moufang Sets With Root Groups of Even Order

Baumeister

Grüninger

2011

Commun. Contemp. Math.

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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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Special Moufang sets with abelian Hua subgroup

Cited by 5 publications

References 5 publications

On cubic action of a rank one group

On cubic action of a rank one group

Trees, contraction groups, and Moufang sets

Finite Zassenhaus Moufang Sets With Root Groups of Even Order

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