arXiv subjects
Publications and source records attributed to Klaus Nielsen.
There is a conjecture, usually attributed to J. G. Thompson, that a finite simple nonabelian group $G$ has a conjugacy class $\Psi$ of covering number 2; i.e $G = \Psi^2$. We show that the commutator subgroup $\Omega(V,Q)$ of the orthogonal group $O(V,Q)$ of a hyperbolic quadratic vector space $(V, Q)$ over a field $K$ has a conjugacy class $\Psi$ such that $\Omega(V,Q) = \Psi^2 \cup -\Psi^2$.
A cyclic square matrix (and its conjugacy class) C over a field K is called (m,k)-cyclic if it has a decomposition $C = A \oplus B$, where $\dim A = m, \dim B = k$ and $m, k \ne 0$. It is shown that the product of two nonsingular (m,k)-cyclic conjugacy classes $\Omega$ and $\Psi$ of GL(m+k,K) contains all nonscalar matrices P in GL(m+k,K) with determinant $\det P = \det \Omega \Psi$.
We show that the symplectic groups Sp(2n, K) over a field $K$ are 5-reflectional if $2n \ge 4$. If $n$ is even and $(2n, K) \ne (4, {\rm GF}(3))$ they are even 4-reflectional.
We call a conjugacy class of the symplectic group Sp$(2n, K)$ over a field $K$ strictly hyperbolic if its minimal polynomial is of the form $q(x) q^*(x)$, where the polynomial $q(x)$ is prime to its reciprocal $q^*(x) := x^n q(x^{-1})$. It is shown that the product of 2 cyclic, strictly hyperbolic conjugacy classes of Sp$(2n, K)$ contains all nonscalar elements of Sp$(2n, K)$. It follows that the projective symplectic group has a conjugacy class of covering number 2, i.e. PSp$(2n,K) = Ω^2$ for some conjugacy class $Ω$ of PSp$(2n,K)$. This verifies a conjecture of J. G. Thompson in the special case of a (finite) projective symplectic group.
It is shown that the product of two nonscalar conjugacy classes of the special linear group SL$(n,K)$ contains matrices of arbitrary trace if $n \ge 4$ and $K$ is an abitrary field or $n=3$ and $K$ is finite.
We classify bireflectional elements (products of 2 involutions) in symplectic groups Sp$(2n, K)$ over a field $K$. We also classify rev ersible elements (elements conjugate to their inverses) and bireflectional elements in finite projective symplectic groups PSp$(2n,q)$.
We classify the bireflections (products of 2 involutions) in the commutator subgroup G an orthogonal group O(V) over a finite field GF(q) of characteristic not 2. We show that every element of G is a bireflection if it is reversible (conjugate to its inverse in G), except when $q \equiv 3 \mod 4, \dim V \equiv 2 \mod 4$ and $V$ is hyperbolic. We also classify the reversible elements of G.
Let $O(p,q)$ be the orthogonal groups of signature $(p,q)$ over the reals. It is shown that an element of the commutator subgroup $O(p,q)'$ of $O(p,q)$ is bireflectional (product of 2 involutions in $O(p,q)'$) if and only if it is reversible (conjugate to its inverse). Moreover, the bireflectional elements of $O(p, q)'$ are classified.
It is shown that a transformation in the special orthogonal group SO(V) of a nondefective quadratic space over a field K is bireflectional (product of 2 involutions) if and only if it is reversible (conjugate to its inverse). Furthermore, all elements of SO(V) are bireflectional if and only if dim V is odd or divisible by 4, or V is a hyperbolic plane over GF(2) or GF(3).