A note on pseudo-reflections (Q580474)

Let V be a finite dimensional vector space and G a group of automorphisms of V generated by pseudo-reflections, i.e. linear transformations \(T: V\to V\) satisfying \(rank(T-id_ V)=1\). Suppose V carries a nondegenerate bilinear form invariant under G. The author shows that if the elements of G have no common fixed vector, then for every linear transformation \(T: V\to V\), there exists \(g\in G\) such that g-T is invertible.