Normal p-subgroups of solvable linear groups

In his paper [8], N. Itô gives an elegant proof that the Sylow p-group of a finite solvable linear group of degree n over the field of complex numbers is necessarily normal if p > n+1. Moreover he shows that this bound on p is the best possible when p is a Fermat prime (i.e. a prime of the form 2sk + 1) but that the bound may be improved to p > n when p is not a Fermat prime.