A bound for the nilpotency of a group of self homotopy equivalences

Let E-Omega(X) be the group of homotopy classes of self-homotopy equivalences of X such that Omega f similar or equal to 1d(Omega X). We prove that epsilon(Omega)(X) is a nilpotent group and that nil epsilon(Omega)(X) less than or equal to cat(X) - 1.