A Counterexample to a Conjecture of Moreto

Abstract: As an extension of the It^o-Michler theorem to the nonabelian or nonnormal Sylow subgroups, A. Moreto raised the following conjecture: let G be a finite group and P a Sylow p-subgroup of G. Then the number of irreducible characters of G whose degrees are divisible by p is not less than the length of any subgroup chain between N_G(P) and G. In this short note, we give a counterexample to this conjecture.