Conjugacy classes with covering number two in hyperbolic orthogonal groups

Abstract

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 $\Omega(V,Q) = \Psi^2 \cup -\Psi^2$.