On normal subgroups and simple modules with a given vertex in a \(p\)-solvable group.
type: Item
instance of: scholarly article
title: On normal subgroups and simple modules with a given vertex in a \(p\)-solvable group.
zbMATH Open document ID: 1120.20015
published in: Journal of Algebra
publication date: 2007-04-18T00:00:00Z
review text: Let \(F\) be an algebraically closed field of characteristic \(p>0\), and let \(Q\) be a \(p\)-subgroup of a finite \(p\)-solvable group \(G\). For a normal subgroup \(N\) of \(G\) and a simple \(FN\)-module \(M\), the set of isomorphism classes of simple \(FG\)-modules with vertex \(Q\) lying over \(M\) is denoted by \(\text{IRR}^Q(FG|M)\). The author proves that \(|\text{IRR}^Q(FG|M)|=|\bigcup_{x\in G}\text{IRR}^Q(FNN_G(Q)|M^*)|\). This generalizes results of the author [Arch. Math. 84, No. 5, 385-391 (2005; Zbl 1074.20005)], of \textit{T. Okuyama} (unpublished) and of \textit{I. M. Isaacs} and \textit{G. Navarro} [J. Algebra 177, No. 2, 339-366 (1995; Zbl 0843.20008)]. The author's result is deduced from a more general theorem where the prime \(p\) is replaced by \(\pi\), a set of primes.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 5144420
zbMATH Keywords: vertices
zbMATH Keywords: characters
zbMATH Keywords: nuclei
zbMATH Keywords: simple modules
zbMATH Keywords: finite \(p\)-solvable groups
author: Abdellatif Laradji
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.05.018
OpenAlex ID: W1986741256
cites work: Q4132720
cites work: Q4103728
cites work: Characters of \(\pi\)-separable groups
cites work: Fong characters in \(\pi\)-separable groups
cites work: Weights and vertices for characters of \(\pi\)-separable groups
cites work: Relative \(\pi\)-blocks of \(\pi\)-separable groups. II
cites work: Vertices of simple modules and normal subgroups of \(p\)-solvable groups
cites work: On simple modules and normal subgroups of \(p\)-solvable groups.
cites work: On normal subgroups and height zero Brauer characters in a \(p\)-solvable group.
DOI: 10.1016/J.JALGEBRA.2006.05.018
description: scientific article; zbMATH DE number 5144420