Sylow 2-subgroups of rational solvable groups.
type: Item
instance of: scholarly article
title: Sylow 2-subgroups of rational solvable groups.
zbMATH Open document ID: 1270.20004
author: I. M. Isaacs
author: Gabriel Navarro
published in: Mathematische Zeitschrift
publication date: 2012-12-19T00:00:00Z
review text: By definition a finite group \(G\) is called rational if every element \(g\) in \(G\) is conjugate to all elements \(g^m\) where \(m\) is coprime to \(|G|\). This is equivalent to the fact that every irreducible character of \(G\) is rational valued. An old conjecture stated by \textit{D. Kletzing}, [Structure and representations of \(\mathbf Q\)-groups. Lect. Notes Math. 1084. Berlin: Springer-Verlag (1984; Zbl 0546.20005)] proposes that if \(G\) is a rational group, then a Sylow 2-subgroup of \(G\) should be a rational group, too. This conjecture appears again in The Kourovka notebook [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka note book. Unsolved problems in group theory. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2006; Zbl 1084.20001)] as question number 16.36. In the paper under review the authors prove that the conjecture is false by finding two counterexamples of order 1536. As a matter of fact they located these two groups using both group theory computer packages GAP and Magma and showed that they are rational groups but their Sylow 2-subgroups have 53 irreducible characters with only 2 non-rational irreducible characters.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6116845
zbMATH Keywords: finite rational groups
zbMATH Keywords: solvable groups
zbMATH Keywords: Sylow 2-subgroups
zbMATH Keywords: 2-rational groups
zbMATH Keywords: rational valued irreducible characters
reviewed by: M. R. Darafsheh
describes a project that uses: GAP
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s00209-011-0965-9
OpenAlex ID: W1966074414
cites work: Rational defect groups and 2-rational characters
cites work: Characters of Solvable and Symplectic Groups
cites work: Q3413656
cites work: Conjugacy in groups of upper triangular matrices
cites work: Structure and representations of Q-groups
DOI: 10.1007/S00209-011-0965-9
description: scientific article; zbMATH DE number 6116845