Generalized permutation characters of solvable groups
type: Item
instance of: scholarly article
title: Generalized permutation characters of solvable groups
zbMATH Open document ID: 0596.20006
DOI: 10.1016/0021-8693(85)90170-X
author: Alan E. Parks
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let \(\chi\) be an irreducible complex character of the finite group G and \(\sigma\in Aut(G)\). Write \(sp(\chi)\) for the sum of the distinct Galois conjugates of \(\chi\) ; here a Galois conjugate \(\chi^{\sigma}\) is the irreducible character of G determined by \((\chi^{\sigma})(g) = (\chi(g))^{\sigma}\). Define P(G) as the set of integral linear combinations of ordinary permutation characters of G. The following is a theorem by E. Artin: Let \(\chi\in Irr(G)\). Then \(| G| \cdot sp(\chi)\in P(G)\). Therefore there is a least positive integer \(n(\chi)\) such that \(n(\chi)sp(\chi)\in P(G).\) The main bulk of this paper is devoted to the proof of the following theorems. (A) Let G be solvable and \(\chi\in Irr(G)\). Then the odd part of \(n(\chi)\) divides \(\chi(1).\) (B) Let G be solvable and \(\chi\in Irr(G)\). Assume \(\chi\) (1) is a power of 2. Then n(\(\chi)\) divides \(\chi(1).\) [In the author's paper in J. Algebra 93, 439-444 (1985; Zbl 0594.20006), it was shown that, for every \(2^ a\), \(a\geq 1\), there is a solvable group G and some \(\chi\in Irr(G)\) such that \(\chi\) (1) is odd and \(2^ a\) divides \(n(\chi)\).] The paper under review contains several substantial results on supersolvable groups, symplectic modules, cyclic normal subgroups, canonical characters, permutation characters, which makes it an important basic one to study.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 3959742
zbMATH Keywords: irreducible complex character
zbMATH Keywords: Galois conjugates
zbMATH Keywords: permutation characters
zbMATH Keywords: supersolvable groups
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
cites work: Q5340151
cites work: Q5662090
cites work: Tensor induction of generalized characters and permutation characters
cites work: Q4124997
cites work: Endliche Gruppen I
cites work: Characters of Solvable and Symplectic Groups
cites work: Q4103728
cites work: Primitive characters, normal subgroups, and M-groups
cites work: Q5569508
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description: scientific article; zbMATH DE number 3959742