Character degrees, derived length and Sylow normalizers
Let \(G\) be a finite solvable group and let \(\pi\) be a set of primes. Let \(\text{cd}_{\pi'}(G)\) denote the set of the degrees of the irreducible complex characters of \(G\) which are relatively prime to all the primes in \(\pi\). Let \(H\) be a Hall \(\pi\)-subgroup of \(G\). Assume that the irreducible characters of \(G\) of \(\pi'\)-degree are monomial. Then it is proved that the derived length of \(N_G(H)/H'\) is less than or equal to the cardinality of \(\text{cd}_{\pi'}(G)\). Here \(N_G(H)\) denotes the normalizer of \(H\) in \(G\) and \(H'\) denotes the derived subgroup of \(H\). The proof uses a character correspondence due to \textit{I. M. Isaacs} [Proc. Am. Math. Soc. 109, No. 3, 647-651 (1990; Zbl 0706.20010)] to be able to follow an argument originally used by Taketa. (Also submitted to MR).
scientific article; zbMATH DE number 1028216