Erich W. Ellers

Decomposition of orthogonal, symplectic, and unitary isometries into simple isometries

Length theorems are of great importance for the characterization of classical groups. The axiom of the three reflections plays a central role in the systems of axioms of F. BACHMANN [2], H. KARZEL [12], R. LINGENBERG [14], and E. SpERNER [17]. These authors characterize motion groups of planes. For characterizations in higher dimensions the length theorems appear to be the right tool to replace the axiom of the three reflections.

Covering numbers for Chevalley groups

LetG be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank ofG, i.e. we give a constantd such that for every noncentral conjugacy classC ofG we haveCrd=G, wherer=rankG.

Factorization of transformations over a local ring

Abstract Let V be a finitely generated free module over a local ring R , and π an invertible linear transformation for V . Then π is a product of simple mappings. In addition, the determinant of each simple factor but one can be chosen to be any given unit in R . The smallest number of factors needed is not greater than r +1, where r is the codimension of the vector space that is associated with the module of vectors fixed under π.

Products of two involutory matrices over skewfields

Abstract If a simple transformation σ is a product of two involutions, then σ is a reflection or a transvection. This property is still true for matrices over skewfields. It will be used to show that a criterion for the decomposability of a matrix into two involutions, which is known for matrices over commutative fields, is no longer true if the entries of the matrix are taken from a skewfield. Another consequence is that the special linear group of a vector space over the field K is not bireflectional if K is not commutative.

Gauss decomposition with prescribed semisimple part in chevalley groups iii: Finite twisted groups

Continuing the investigations of [EG] and [EGII], we shall show that Theorem 1 below is also valid for twisted CheMLley groups over finite fields. Let G be such a group. Here we consider only groups G 2 ZF/2, where z is a universal Chevalley group over s finite field K, F is an automorphism of z, and Z is a subgroup of G contained in the center z(@). Suppose B = HU is a Bore1 subgroup of G. Let I? be a group generated by G and some element o normalizing G in I? and acting on G as diagonal automorphism.

Products of quasireflections and transvections over local rings

Let R be a commutative local ring and M a free R-module of rank n. The module M is endowed with a metric structure given by a sesquilinear form or by a quadratic form. We show, every isometry π of M is a product of quasireflections or transvections. We determine the minimal number of factors needed in any factorization of π if the path of π is a subspace. For all other isometries we obtain only an upper bound.

Length theorems for the general linear group of a module over a local ring

ERICH W. ELLERS and HUBERTA LAUSCH(Received 10 November 1986; revised 5 November 1987)Communicated by R. LidlAbstractLet R be a not necessarily commutative local ring, M a free ij-module, and ir 6 GL(M) suchthat B{n) = im(7 —r 1 ) is a subspace of M. The —n o\ ir • • -atp, wher Oie are simple mappingsof given types, p is a simple mapping B(CTJ, an)d B(p) are subspaces, and t < dimB(7r).1980 Mathematics subject classification (Amer. Math. Soc.) (198 Revision):5 primary 15 A 23;secondary 20 H 15.Keywords and phrases: local ring, module, general linear group, factorization, simple mapping.

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.

Products of λ-transvections in Sp(2n,q)

Abstract Every transvection A in the symplectic group Sp(2 n , q ), q a power of an odd prime, is a product of transvections T i , where all T i are in the same conjugacy class C of transvections. For each transformation A we determine the length of A , i.e., we find the least number of transvections in C required to express A .

Products of half-turns

Abstract LetV be a vector space of any dimension andO+ (V) the proper orthogonal group ofV. We show, if π eO+ (V), then π is a product of half-turns, and we determine the minimal number of half-turns which is needed to express π.