Let G be a finite group. Fix
a prime integer p and let e be the largest integer such that pe divides the
degree of some irreducible Brauer character of G with respect to the same
prime p. The primary object of this paper is to obtain information about the
structure of Sylow p-subgroups of a finite p-solvable group G in knowledge of
e.
As applications, we obtain a bound for the derived length of the factor group of a
solvable group G relative to its unique maximal normal p-subgroup in terms of the
arithmetic structure of its Brauer character degrees and a bound for the derived
length of the factor group of G relative to its Fitting subgroup in terms of the
maximal integer e when p runs through the prime divisors of the order of
G.
