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

8 pages