2-length and rational characters of odd degree.
type: Item
instance of: scholarly article
title: 2-length and rational characters of odd degree.
zbMATH Open document ID: 1227.20003
published in: Archiv der Mathematik
publication date: 2011-05-11T00:00:00Z
review text: Let \(G\) be a finite solvable group of 2-length \(l\). \textit{G. Navarro} and \textit{P. H. Tiep} [Trans. Am. Math. Soc. 360, No. 5, 2443-2465 (2008; Zbl 1137.20009)] have shown that \(l\) is less than the number of rational irreducible characters of odd degree. In the paper under review the author improves this linear upper bound to a logarithmic bound by proving that the number of rational irreducible characters of \(G\) of odd degree is at least \(2^l\). He also provides examples showing that for every \(l\) this bound is best possible. Moreover, the main result together with the above-mentioned work of Navarro and Tiep easily implies the following group theoretic consequence: If \(P\) is a Sylow 2-subgroup of \(G\), then the number of \(N_G(P)\)-orbits on \(P/\Phi(P)\) is at least \(2^l\). -- In this reviewer's opinion, it would be interesting to (1) find a purely group theoretical proof for this fact and (2) to extend it to odd primes.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5886764
zbMATH Keywords: finite solvable groups
zbMATH Keywords: 2-lengths
zbMATH Keywords: numbers of rational irreducible characters
zbMATH Keywords: numbers of orbits
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: numbers of irreducible characters
zbMATH Keywords: rational valued characters
author: Joan Tent
MaRDI profile type: Publication
full work available at URL: s00013-011-0229-2
OpenAlex ID: W1963493905
cites work: Correspondences between constituents of projective characters.
cites work: Real‐Valued Characters of Solvable Groups
cites work: Q3413656
cites work: Rational irreducible characters and rational conjugacy classes in finite groups
DOI: 10.1007/S00013-011-0229-2
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description: scientific article; zbMATH DE number 5886764