Defect groups and character heights in blocks of solvable groups. II (Q790238)

[For Part I cf. the second author, ibid. 72, 183-209 (1981; Zbl 0472.20004)]. The main result is that if N is a normal subgroup of a group G, G/N is solvable, and N has an irreducible character \(\phi\) so that the prime p does not divide \(\chi\) (1)/\(\phi\) (1) whenever \(\chi\) is an irreducible constituent of \(\phi^ G\), then the Sylow p-subgroups of G/N are abelian. As a consequence one obtains a case of Brauer's conjecture. To wit, if \(G/O_{p'}(G)\) is solvable and if every character in some p- block B of G is of height zero, then a defect group of B is abelian. More generally, the method yields some bounds on derived length of Sylow subgroups in terms of character degrees and heights.