Klaus Nielsen

Products of Involutions in O + (V)

Abstract Let V be a finite-dimensional vector space over a commutative field of characteristic distinct from 2. Let V carry a symmetric nondegenerate bilinear form. The special orthogonal group is O + ( V ):={ πϵO ( V ): det π = 1}. Main result: An element π ∈ O + ( V ) is a product of two involutions in O + ( V ) ifand only if dim V ≢ 2 mod 4 or an orthogonal decomposition of V into orthogonally indecomposable π-modules contains a π-module of odd dimension.

On products of two involutions in the orthogonal group of a vector space

Let V be a finite-dimentional vector space over a commutative field of characteristic distinct from 2. Let V carry a symmetric nondegenerate bilinear form. Results: (A) Let π = ρσ, where π, ρ, σ ∈ O(V) and ρ, σ are involutions. There exists an orthogonal decomposition of V into orthogonally indecomposable π-modules which are simultaneously invariant under ρ and σ. (B) Let π ∈ O(V).One can find involutions ρ, σ ∈ O(V) such that π = ρσ and B(π) = B(ρ) + B(σ) holds if and only if an orthogonal decomposition of V into orthogonally indecomposable π-modules does not contain a term whose minimum polynomial is (x−1)α where α is even.

On bireflectionality and trireflectionality of orthogonal groups

Let V be a finite-dimensional vector space over a field of characteristic different from two. Let V carry a symmetric bilinearform. We show that the weak orthogonal group O∗(V) is always trireflectional, and is bireflectional if and only if the index of the form or the dimension of the radical is at most 1.

A matrix-decomposition theorem for GLn(K)

Abstract Given an arbitrary commutative field K , n∈ N ⩾3 and two monic polynomials q and r over K of degree n−1 and n such that q (0)≠0≠ r (0). We prove that any non-scalar invertible n×n matrix M can be written as a product of two matrices A and B , where the minimum polynomial of A is divisible by q and B is cyclic with minimum polynomial r . This result yields that the Thompson conjecture is true for PSL n ( F 3 ) , n∈ N ⩾3 , and PSL 2n+1 ( F 2 ) , n∈ N . If G is such a group, then G has a conjugacy class Ω such that G=Ω 2 . In particular each element of G is a commutator.

Products of symmetries in unitary groups

Abstract Given a regular − -hermitian form on a finite-dimensional vector space V over a commutative field K of characteristic ≠2 such that the norm on K is surjective onto the fixed field of − (this is true whenever K is finite). Call an element σ of the unitary group a symmetry if σ 2 = 1 and the negative space of σ is 1-dimensional. If π is unitary and det π ∈ {1, − 1}, we prove that π is a product of symmetries (with a few exceptions when K = GF 9 and dim V = 2) and we find the minimal number of factors in such a product.

k-fold anti-invariant subspaces of a linear mapping

Abstract Let π be a linear bijection on a finite-dimensional vector space and k ⩾0 an integer. A subspace T is called a k -fold anti-invariant subspace if T + Tπ +⋯+ Tπ k = T ⊕ Tπ ⊕⋯⊕ Tπ k . We find the maximal dimension of such subspaces T .

The product of two quadratic matrices

Abstract Let p=(x−β)(x−β −1 )∈K[x] where β 2 ≠β −2 and let V be a finite-dimensional vector space over the field K . A linear mapping M:V→V is called quadratic if p(M)=0 . We characterize products of two quadratic linear mappings.