The q-parts of degrees of Brauer characters of solvable groups (Q1823308)

Let G be a finite solvable group, and let p and q be different primes. The authors show that, if \(q^{e+1}\) does not divide the degree of any irreducible Brauer character of G in characteristic p then the derived length of a Sylow p-subgroup of G is at most \(4e+3\). Similarly, if \(p^{e+1}\) does not divide the degree of any irreducible Brauer character of G in characteristic p then the p-rank of \(G/O_ p(G)\) is at most pe/(p-1); here the p-rank of a finite solvable group H is the largest integer r such that \(p^ r\) is the order of a p-chief factor of H. Some consequences and related results are also proved.