Groups whose non-linear irreducible characters are rational valued.
type: Item
instance of: scholarly article
title: Groups whose non-linear irreducible characters are rational valued.
zbMATH Open document ID: 1198.20008
author: B. E. Eshmatov
published in: Archiv der Mathematik
publication date: 2010-06-04T00:00:00Z
review text: Let \(G\) be a finite group. \(G\) is called a rational group or \(\mathbb{Q}\)-group if all irreducible characters of \(G\) are rational valued, and \(G\) is called a \(\mathbb{Q}_1\)-group if all nonlinear irreducible characters of \(G\) are rational valued. The paper under review presents examples of nonabelian \(\mathbb{Q}_1\)-groups which are not \(\mathbb{Q}\)-groups and also presents some results on the structure of \(\mathbb{Q}_1\)-groups. For example, if \(G\) is nonabelian, then \(G\) is a \(\mathbb{Q}_1\)-group if and only if \(\chi(g)\in\mathbb{Q}\) for all \(\chi\in\text{Irr}(G)\) and all \(g\in V(G)\). (Here \(V(G)\) denotes the vanishing-off subgroup of \(G\) which is generated by all \(g\in G\) such that there is a nonlinear \(\chi\in\text{Irr}(G)\) with \(\chi(g)\neq 0\).) Combining this with a result of M. Lewis that \(G=V(G)\) for nonsolvable groups shows that the notions of \(\mathbb{Q}\)-group and \(\mathbb{Q}_1\)-group are the same for nonsolvable groups.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 5716891
zbMATH Keywords: finite groups
zbMATH Keywords: rational characters
zbMATH Keywords: vanishing-off subgroups
zbMATH Keywords: irreducible characters
MaRDI profile type: Publication
full work available at URL: s00013-010-0110-8
OpenAlex ID: W1989764518
cites work: Q4220407
cites work: 2-Frobenius \(\mathbb{Q}\)-groups.
cites work: On finite rational groups and related topics
cites work: Q4103728
cites work: The vanishing-off subgroup.
cites work: Structure and representations of Q-groups
cites work: Structure of solvable rational groups
DOI: 10.1007/S00013-010-0110-8
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description: scientific article; zbMATH DE number 5716891