Sylow subgroups and the number of irreducible characters of degrees divisible by a prime p

Abstract

Let be a finite group and be a prime. We establish an upper bound for the derived length of a Sylow ‐subgroup of in terms of the number of irreducible characters of whose degrees are divisible by . We also prove that if is a ‐block of a finite ‐solvable group with defect group , then the derived length of is at most one more than the number of ordinary irreducible characters of positive height in .