On a theorem of J. A. Green
type: Item
instance of: scholarly article
title: On a theorem of J. A. Green
zbMATH Open document ID: 0919.20005
author: Kenichi Yamauchi
published in: Journal of Algebra
publication date: 1999-07-06T00:00:00Z
review text: Brauer's famous induction theorem states that each irreducible complex character of a finite group \(G\) is expressible as an integral linear combination of linear characters induced from elementary subgroups. (An elementary subgroup is a direct product of a \(p\)-subgroup with a cyclic \(p'\)-subgroup, for some prime \(p\).) \textit{J. A. Green} [Proc. Camb. Philos. Soc. 51, 237-239 (1954; Zbl 0057.02102)] showed conversely that if \(\mathcal H\) is a family of subgroups of \(G\) with the property that each irreducible character of \(G\) is expressible as an integral linear combination of characters induced from subgroups in \(\mathcal H\), then each elementary subgroup of \(G\) is contained in a conjugate of some subgroup in \(\mathcal H\). The author of the present paper offers a new proof of Green's theorem, without using the formula for induced characters. The new ingredient of his proof is the use of the characteristic class functions of \(G\), that is, the functions that take the value 1 on all elements of a given conjugacy class, and 0 on all other elements. Examination of Section 15 F of \textit{C. W. Curtis} and \textit{I. Reiner}, Methods of Representation Theory, Volume 1 [John Wiley, New York (1981; Zbl 0469.20001)], suggests that the author's proof differs little in its details from existing proofs, and retains the usual final contradiction, that a certain prime divides the value of the principal character on some element.
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1247623
zbMATH Keywords: Brauer induction theorem
zbMATH Keywords: elementary subgroups
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: finite groups
zbMATH Keywords: linear combinations of linear characters
zbMATH Keywords: characteristic class functions
reviewed by: Rod Gow
MaRDI profile type: Publication
full work available at URL: jabr.1998.7552
OpenAlex ID: W1994744732
cites work: A characterization of the characters of groups of finite order
cites work: Q5340151
cites work: Q5829823
cites work: Q4103728
cites work: On isomorphisms of a Brauer character ring onto another
cites work: On automorphisms of a character ring
DOI: 10.1006/JABR.1998.7552
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description: scientific article; zbMATH DE number 1247623