\(p\)-Steinberg characters of finite simple groups
type: Item
instance of: scholarly article
title: \(p\)-Steinberg characters of finite simple groups
zbMATH Open document ID: 0874.20009
published in: Journal of Algebra
publication date: 1997-11-04T00:00:00Z
review text: Let \(p\) be a prime number and \(G\) be a finite group such that \(p\mid|G|\). A \(p\)-Steinberg character of \(G\) is an irreducible complex character \(\chi\) of \(G\) such that \(\chi(x)=\pm|C_G(x)|_p\) for every \(p'\)-element \(x\) of \(G\), where \(n_p\) denotes the \(p\)-part of the natural number \(n\). The notion of a \(p\)-Steinberg character was introduced by \textit{W. Feit} [in Linear algebraic groups and their representations, Los Angeles 1992, Contemp. Math. 153, 1-9 (1993; Zbl 0823.20013)] in which the following conjecture is made: Let \(p\) be a prime number and \(G\) a finite simple group such that \(p\mid|G|\). If \(G\) has a \(p\)-Steinberg character, then does it follow that \(G\) is isomorphic to a simple group of Lie type defined in characteristic \(p\)? Feit suggests using the classification of finite simple groups the above question may be answered affirmatively. Using the character tables of the sporadic simple groups [in \textit{J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker} and \textit{R. A. Wilson}, Atlas of finite groups (Clarendon, Oxford, 1985; Zbl 0568.20001)] it is easy to see that none of the 26 sporadic simple groups has a \(p\)-Steinberg character. For the alternating and the projective special linear groups, the reviewer proved W. Feit's conjecture [in J. Algebra 181, No. 1, 196-206 (1996; Zbl 0851.20008)]. In the paper under review the author proves the above mentioned conjecture of Feit for the rest of the finite simple groups.
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 1014045
zbMATH Keywords: \(p\)-Steinberg characters
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: simple groups of Lie type
zbMATH Keywords: classification of finite simple groups
zbMATH Keywords: character tables
zbMATH Keywords: sporadic simple groups
author: Pham Huu Tiep
reviewed by: M. R. Darafsheh
MaRDI profile type: Publication
full work available at URL: jabr.1997.6791
OpenAlex ID: W1967809792
cites work: Q3684278
cites work: \(p\)-Steinberg characters of alternating and projective special linear groups
cites work: Q4278076
cites work: On the minimal degrees of projective representations of the finite Chevalley groups
cites work: On the minimal degrees of projective representations of the finite Chevalley groups. II
cites work: Minimal characters of the finite classical groups
DOI: 10.1006/JABR.1997.6791
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description: scientific article; zbMATH DE number 1014045