Characters of groups with quotients of odd order
type: Item
instance of: scholarly article
title: Characters of groups with quotients of odd order
zbMATH Open document ID: 0567.20003
DOI: 10.1016/0021-8693(85)90037-7
author: I. M. Richards
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: The following theorem is proved: Let G be a group of order g. Let \(H\triangleleft G\), \([G:H]=m\), m being odd. Let \(\tau \in Gal({\mathbb{Q}}_ g/{\mathbb{Q}})\) be such that \(q^{\tau}=\bar q\), for all \(q\in {\mathbb{Q}}_ m\). Let \(\psi\in Irr(H)\) be fixed by \(\tau\). There exists a unique irreducible constituent of \(\psi^ G\), \(\chi\), such that \(\chi\) is fixed by \(\tau\). Further, \({\mathbb{Q}}(\chi)\subseteq {\mathbb{Q}}(\psi^ G)\). There are four corollaries. A remarkable one is this. Corollary. If every simple group of even order has an irreducible rational character of odd degree, other than the trivial, then the same is true for all groups of even order.
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 3904813
zbMATH Keywords: irreducible constituent
zbMATH Keywords: irreducible rational character
zbMATH Keywords: groups of even order
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90037-7
OpenAlex ID: W2064913952
cites work: Real‐Valued Characters of Solvable Groups
cites work: Q4103728
description: scientific article; zbMATH DE number 3904813