Gernot Stroth

Quadratic GF 2-modules for sporadic simple groups and alternating groups

Let G he a finite group and V a faithful, finite dimensional GF 2 G-module. We call E ≤G quadratic,if[V,E,E]=1 and |E|≥4, and we say V is quadratic, if G is generated by its quadratic subgroups. It is an easy exercise, but important for what follows, to show that E has to be elementary abelian. Elementary abelian groups acting quadratically come up quite naturally in weak-closure arguments, pushing-up problems and in problems dealing with parabolic systems. As a successor of [MS] we treat the case that G is a sporadic simple group or an alternating group.

Groups and Geometries

As groups are just the mathematical way to investigate symmetries, it is not surprising that a significant number of problems from various areas of mathematics can be translated into specialized problems about permutation groups, linear groups, algebraic groups, and so on. In order to go about solving these problems a good understanding of the finite and algebraic groups, especially the simple ones, is necessary. Examples of this procedure can be found in questions arising from algebraic geometry, in applications to the study of algebraic curves, in communication theory, in arithmetic groups, model theory, computational algebra and random walks in Markov theory. Hence it is important to improve our understanding of groups in order to be able to answer the questions raised by all these areas of application.

On Chevalley-groups acting on projective planes

In [IO] Hcring introduced the concept of a strongly irreducible collineation group. Let n be a projective plane, K a collineation group of n; K is said to be strongly irreducible iff K fixes no points, lines triangles. or subplanes of n. If K is a finite group acting strongly irreducible on 71 generated by perspectivities, then Hering shows in ] 10 ] that K is either an extension of a 3-group with a subgroup of the automorphismgroup or there is a normal subgroup G in K, G a nonabelian finite simple group, such that K < Aut(G). The aim of this paper is to prove the following theorem. ,

One Node Extensions of Buildings

A chamber system τ = (τ, (ρ i)i ∈ I) over some index set I is a set τ of chambers together with partitions ρ i, i ∈ I, of τ. If J ⊆ I, then ΔJ is the join of all partitions ρ j, j ∈ J. If c ∈ τ and J ⊆ I, then ΔJ(c) is the element of ρ J containing c. Notice that ΔJ(c) is again a chamber system over J with partitions ρ J, j ∈ J, restricted to ΔJ(c). τ is called connected iff ΔJ(c) = {τ} for some c ∈ τ. The permutation group G of τ is an automorphism group of τ if it respects all the partitions ρ i, i ∈ I. Let rank τ = |I|. A chamber system is called tight if there is some i ∈ I with {τ} = ΔI–{i}(c).

Strong quadratic modules

For a Chevalley groupG over a field of characteristic 2 we determine all irreducible modulesV overGF(2) such that [V, R, Q]=0, whereR is a long root group andQ=Z2(O2(NG(R))). As a corollary we obtain a classification of those irreducible modules admitting a quadratic fours groupE which intersect a long root group nontrivially but is not contained in such a group.

Check Character Systems Using Chevalley Groups

We show that the Sylow 2-subgroups of nearly all Chevalley groups in even characteristic allow the definition of a check-character-system which detects all single and the most important double errors.

Classification of certain types of tilde geometries

We show that a certain class of diagram geometries called tilde geometries of symplectic type is simply connected. Here we prove that the corresponding amalgam is uniquely determined. The result then follows from Ivanov and Shpectorov (Geom. Dedicata45 (1993), 1–23).

Modified Steinberg Relations for the Group J 4

Let Φ be a root system of type D 5 and let Δ ⊆ Φ be a fundamental system of roots, which we label with the integers 1,2,…, 5. For each i ∈ Δ there is a unique involution αi in the corresponding root group of the Chevalley group D 5(2). The opposite root corresponds to an involution we will call α–i . We let w i = α i α–i for each i ∈ Δ and assume that the labels are chosen so that (w 1 w 2)3 = (w 2 w 3)3 = (w 3 w 4)3 = (w 3 w 5)3 = 1. We will denote commutators of the elements α1,…, α5 with multiple subscripts, i.e. α12 = [α1, α2], α123 = [α12, α3] etc. The elements α1,…, α5, w 5,…,w 5 generate D 5(2) and satisfy the Steinberg relations [3, p. 190], which may be used to define D 5(2) abstractly.

Einige Gruppen vom Charakteristik 2-Typ

Eine endliche Gruppe G hei& vom Charakteristik 2-Typ, falls fiir jede Involution t in G stets F*(C,(t)) eine 2-Gruppe ist. Es ist wahrscheinlich, da0 sich das Problem alle einfachen Gruppen G zu bestimmen in naher Zukunft darauf reduziert, da0 G vom Charakteristik 2-Typ ist. In diesem Fall scheint die Kennzeichnung von G durch den Zentralisator eines Elementes p von der Ordnung drei sinnvoll. Ein wesentlicher S&M bei solchen Kennzeichnungen ist die Bestimmung gewisser 2-lokaler Untergruppen L, die p enthalten. Da p auf O,(L) operiert kann man gewisse Aussagen i.iber die Struktur von O,(L) machen. Besonders interessanti ist der Fall, daD L der Zentralisator einer Involution ist. Das legt das Problem nahe, alie bekannten einfachen Gruppen G vom Charakteristik 2-Typ durch die Struktur von O&,(t)) einer Involution t zu kennzeichnen. In fast allen bekannten einfachen Gruppen G vom Charakteristik 2-Typ gibt es eine Involution t, so daB F*(C,(t)) eine extraspezielle Gruppe oder eine Verallgemeinerung davon eine sog. semi-extraspezielle Gruppe ist. Der Fall, daf3 G eine Involution t besitzt, so daO F*(C,(t)) eine extraspezielle Gruppe ist, wurde von Aschbacher [2], Smith [17, 181 und Timmesfeld [25] untersucht. In diesen Arbeiten wurde die Struktur von C,(t) so weit bestimmt, da0 es mijglich ist, die Struktur von G zu bestimmen. Es gibt aber such einfache Gruppen vom Charakteristik 2-Typ, z.B. F,(2), die keine Involution t enthalten, so da6 F*(C,(t)) extraspeziell oder semi-extraspeziell ist. Dann gibt es aber stets eine Involution t, so da8 F*(C,(t)) das direkte Produkt einer elementar abelschen Gruppe mit einer extraspeziellen oder semi-extraspeziellen Gruppe ist. Das Ziel dieser Arbeit ist der Beweis des folgenden Satzes.

An odd characterization ofJ 4

Z. Janko recently discovered a finite simple group calledJ4. The purpose of this paper is to classifyJ4 by the structure of the centralizer of an element of order three.