Character degrees, derived length and Sylow normalizers
type: Item
instance of: scholarly article
title: Character degrees, derived length and Sylow normalizers
zbMATH Open document ID: 0881.20004
author: Gabriel Navarro
published in: Archiv der Mathematik
publication date: 1997-07-06T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1028216
zbMATH Keywords: monomial groups
zbMATH Keywords: derived length
zbMATH Keywords: finite solvable groups
zbMATH Keywords: character degrees
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: Hall subgroups
reviewed by: Alexandre Turull
MaRDI profile type: Publication
full work available at URL: s000130050077
OpenAlex ID: W2071953731
DOI: 10.1007/S000130050077
description: scientific article; zbMATH DE number 1028216