Wolfgang Nolte

Bireflectionality of the weak orthogonal and the weak symplectic groups

Let V be a finite-dimensional vector space, Q a quadratic form and fa = f the bilinear form associated with Q. We assume (V,&) is regular. Then the orthogonal group O(V) is bireflectional, i.e., every isometry in O(V) is a product of two involutory isometries in O(V). This has been shown in [8] if the field of scalars K has characteristic distinct from 2 and in [4] and [5] if char K = 2. The latter papers also establish the bireflectionality for the symplectic group Sp( V), again under the assumption that (V,f) is regular and char K = 2. For char K # 2 the symplectic group is not bireflectional (see [31). We shall extend the results just mentioned. We shall drop the assumptions that V is finite dimensional and that (V,f) is regular, i.e., the vector space V may be infinite dimensional and the radical of V may be distinct from zero. We shall use the notation and the concepts in [2]. For every 7c E Hom(V, V) we define F(n)= {UE V;?m=u} and B(n)= (vz u; ZJ E V}. The spaces F(r) and B(n) are called fix and path of z, respectively. The groups O*(V) = {n E O(V); rad VC F(z) and dim B(n) < co} and Sp *( V) = { 71 E Sp( V); rad V c F(n) and dim B(n) < co ) are called the weak orthogonal and the weak symplectic group, respectively.

A Characterization of Pappian Affine Hjelmslev Planes

In [7], Pickert showed that an affine plane is isomorphic to an affine plane over a field if the dual Pappus theorem holds for two fixed pencils of parallel lines (see also [2]). Here we ask whether a similar result is true for an affine Hjelmslev plane Ω. The geometric structure Ω will be defined in such a generality that the algebraic models of affine Hjelmslev planes over commutative local rings (not only those over Hjelmslev rings) are included. We consider two distinguished not-neighbouring pencils V, W of parallel lines and assume the validity of the configuration conditions (DP), (SD). The main result is the following: If the epimorphic image a(Ω) of Ω is not a Fano plane then the incidence structure of Ω is isomorphic to that of an affine Hjelmslev plane Ω(R) over a local ring R. In R the element 2 is invertible. All pencils of parallel lines in Ω correspond to pencils of parallel lines in Ω(R) with the possible exception of those neighbouring V or W.

Erweiterte Spiegelungsgruppen metrischer Vektorräume

The “extended reflection groups” of a metric vector space (V,f) are introduced and defined by a system of generators and a set of defining relations. It can be proved that they are isomorphic to certain subgroups of the orthogonal groups. The main result of the underlying paper is that these groups can be characterized by a few properties among which we mention the validity of the transitivity theorem and the property of “ Δ — intersecting ”. Finally, we obtain a characterization of the (full) groups O*(V,f) in the case dim V⊥≤1.

Metrische Räume Mit Dreiseitverbindbaren Teilräumen II

A summary of the contents is given in part I of this paper which also is published in the Journal of Geometry.Die Sätze sind, in Teil I beginnend, durchnummeriert. Betr. Einleitung, Inhaltsübersicht und Literaturverzeichnis s. Teil I.

A characterization of subgroups of the orthogonal group

LetG be a subgroup of the general linear group GLn(K), where charK ≠ 2. Put Kn =V. AssumeG is generated by the setS of all elements σ inG for which dimV(σ − 1) = 1, and suppose σ2=1V for each σ inS. If {V(σ−1)¦σ∈S} contains a simplex, if − 1V ∈G, and if π inG is a product of dim v(π−1) elements σ inS wheneverV(π−1) is not contained in the kernel ofπ−1, thenG is a subgroup of an orthogonal group.

Metrische Räume mit Dreiseitverbindbaren

This paper presents a system of axioms for n-dimensional metric geometry. For every group satisfying the axioms there exist a group-space and an embedding of into a projective-metric space Ω. We construct an isomorphism of onto a subgroup of a special orthogonal group On+1*(K,f). This group belongs to a metric vector space (V,f) over a field K of characteristic ≠ 2 where dim rad V≦1. The (full) groups on+1*(K,f) are models of the system of axioms.

Pappussche affine Klingenbergebenen

The result in this paper complements that of Mäurer and Nolte in [8]. Here we show: If the epimorphic imageα(Ω) of an affine Klingenberg plane (Ω, α) is a Fano plane then the configuration conditions (DP),(SD) and (¯ p) imply that Ω is isomorphic to the affine Klingenberg plane over a commutative local ringR, where 1+1 is a non-unit.The corresponding result for the case that α(Ω) is not a Fano plane has been proved without using (¯ p) (see [8], Theorem 2).

Darstellung spezieller projektiver Spiegelungsgruppen durch orthogonale Gruppen

For projective reflection groups of a special type we prove the following theorem: Every such group is isomorphic to a group of orthogonal transformations.

Eine projektive Spiegelungsgruppe, welche S-Gruppe ist

Let π be a projective plane with the flag (A,b). Assume that there exist all involutory homologies with centres on b and axis through A. Let G be the collineation group generated by the set S of all these involutory homologies. It is shown that π is a pappian plane if (G,S) is an S - group.

Groups satisfying Scherk's length theorem

We consider subgroupsG of the general linear groupGL(n,K) where charK≠2. IfG is generated by the setS of its simple involutions, if −1v εG, and if Scherks length theorem holds forG, thenG is a subgroup of an orthogonal group.