Let $G$ be a pseudoreflection group and $H$ be a reflecting hyperplane. Let $G_H$ be the subgroup of $G$ consisting of all those elements of $G$ which stabilize the reflecting hyperplane $H$ pointwise. Show that $G_H$ is a cyclic subgroup of $G.$
Today while going through a paper I found this interesting claim which I can't able to prove. What is a generator of $G_H$? Any help in this regard would be greatly appreciated.
Thanks for your time.