\(p\)-Steinberg characters of alternating and projective special linear groups
type: Item
instance of: scholarly article
title: \(p\)-Steinberg characters of alternating and projective special linear groups
zbMATH Open document ID: 0851.20008
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: Every finite group of Lie type in characteristic \(p\) has a Steinberg character, which has value 0 on every \(p\)-singular element, and whose value on every \(p\)-regular element \(x\) is plus or minus the order of the \(p\)-part of the centralizer of \(x\) (in particular, the degree of the character is the order of the Sylow \(p\)-subgroup). Taking these properties as the definition of a \(p\)-Steinberg character of an arbitrary group, W. Feit asks which simple groups have a \(p\)-Steinberg character, and conjectures that all examples are known. In this paper, the author uses the well-developed character theory of the symmetric groups and general linear groups to show that there are no surprises in these cases. In particular, the only \(p\)-Steinberg characters for alternating groups are those arising from the isomorphisms \(A_5\cong L_2(4)\cong L_2(5)\), \(A_6\cong L_2(9)\) and \(A_8\cong L_4(2)\). Similarly, the only \(p\)-Steinberg characters for \(L_n(q)\), where \(p\) is not the defining characteristic, arise from the isomorphisms \(L_2(4)\cong L_2(5)\) and \(L_2(7)\cong L_3(2)\).
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 883755
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: \(p\)-singular elements
zbMATH Keywords: \(p\)-regular elements
zbMATH Keywords: symmetric groups
zbMATH Keywords: general linear groups
zbMATH Keywords: \(p\)-Steinberg characters
zbMATH Keywords: alternating groups
author: M. R. Darafsheh
reviewed by: Robert A. Wilson
MaRDI profile type: Publication
full work available at URL: jabr.1996.0115
OpenAlex ID: W2055375659
DOI: 10.1006/JABR.1996.0115
description: scientific article; zbMATH DE number 883755