Factorizations of characters and a question of Feit
type: Item
instance of: scholarly article
title: Factorizations of characters and a question of Feit
zbMATH Open document ID: 0628.20012
DOI: 10.1016/0021-8693(87)90095-0
author: Pamela A. Ferguson
author: Alexandre Turull
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let G be a finite group, \(\chi\) a complex character of G, \({\mathbb{Q}}\) the rational field, \({\mathbb{Q}}(\chi)={\mathbb{Q}}(\chi (x)|\) \(x\in G)\), and \({\mathbb{Q}}_ m\) the field of mth roots of unity over \({\mathbb{Q}}\). Let f(X) be the smallest positive integer m wth \({\mathbb{Q}}(\chi)\subseteq {\mathbb{Q}}_ m\). The authors give some affirmative answers to the following question of Feit: If \(\chi\) is an irreducible character of G, is it true that G contains an element of order f(\(\chi)\)? The authors use character factorization to arrive to a number of related results including: Theorem A. Let G be a finite group all of whose non- Abelian simple composition factors are minimal simple. Let \(\chi\) be any irreducible character of G. Then G contains an element of order f(\(\chi)\). Theorem B. Let G be a finite group all of whose composition factors are N-groups. Suppose that \(\chi\) is irreducible and m is some positive integer such that \(m| f(\chi)\). Assume at least one of the following holds: (i) 36\(\nmid \chi (1)\); (ii) \(A_ 7\) is not involved in G (as a composition factor if \(\chi\) is quasi-primitive); (iii) 30\(\nmid m\); (iv) \(G'=G\) and \(\chi\) is quasi-primitive. Then G contains an element of order m.
reviewed by: B.G.Basmaji
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4021424
zbMATH Keywords: irreducible character
zbMATH Keywords: character factorization
zbMATH Keywords: simple composition factors
zbMATH Keywords: N-groups
zbMATH Keywords: quasi-primitive
zbMATH Keywords: element of order m
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90095-0
OpenAlex ID: W2037889653
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description: scientific article; zbMATH DE number 4021424