The Loewy layers of principal indecomposable modules
type: Item
instance of: scholarly article
title: The Loewy layers of principal indecomposable modules
zbMATH Open document ID: 0788.20005
DOI: 10.1016/0021-8693(92)90161-E
author: Michael Weidner
published in: Journal of Algebra
publication date: 1993-03-10T00:00:00Z
review text: Let \(G\) be a \(p\)-solvable group, and let \(k\) be the field of \(p\) elements. The author obtains results about the Loewy layers of the principal indecomposable module \(P_ 1\) corresponding to the trivial module \(k\). For \(G\) a \(p\)-group the structure of the Loewy layers of \(P_ 1 = kG\) is completely described by the Jennings polynomial. Here the author gives a generalization of this polynomial to \(p\)-solvable groups where the situation is much more complicated: For the construction of the polynomial a lot of information about the group and its module theory is needed, also the polynomial does not yield the structure of the Loewy layers in a straightforward way. The specific applications given in this paper include a new proof of the celebrated theorem of Gaschütz about \(P_ 1J/P_ 2J\), a new proof of the result of Green-Hill on the occurrence of the chief factors of \(G\) as composition factors of \(P_ 1\), a complete description of the Loewy layers in case \(G\) is of \(p\)- length 1, and a description of the layer \(P_ 2J/P_ 3J\) in case \(G\) is of \(p\)-length 2.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 130535
zbMATH Keywords: Loewy layers
zbMATH Keywords: principal indecomposable module
zbMATH Keywords: \(p\)-group
zbMATH Keywords: Jennings polynomial
zbMATH Keywords: \(p\)-solvable groups
zbMATH Keywords: composition factors
zbMATH Keywords: \(p\)-length
reviewed by: Urs Stammbach
MaRDI profile type: Publication
cites work: Moduln und verschränkte Homomorphismen endlicher Gruppen.
cites work: Pre-Frattini groups.
cites work: Cover-avoid subgroups in finite solvable groups
cites work: Q3934684
cites work: On the loewy series of the group algebra of groups of smallp-Length∗
cites work: Q3998977
full work available at URL: 0021-8693(92)90161-e
OpenAlex ID: W2087528774
description: scientific article; zbMATH DE number 130535