Split chief factors and cohomology
type: Item
instance of: scholarly article
title: Split chief factors and cohomology
zbMATH Open document ID: 0615.20031
DOI: 10.1016/0022-4049(87)90037-5
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let G be a finite group. Let \(k=F_ p\) be the field with p elements. A simple kG-module M is called a split chief factor if there exist normal subgroups N and K of G such that K/N is isomorphic to M and such that G/N splits over K/N. For any given chief series of G, the number s(G,M) of split chief factors in this series is known to be independent of the choice of the series. The principal result generalizes a result of Gaschütz: Theorem. \(s(G,M)=\dim_ DH^ 1(G,M)-\dim_ DH^ 1(G/CM,M)\), where \(D=End_{kG}M\) and \(CM=\ker nel\) of the action of G on M. In conjunction of an older result of the author, one gets a converse of a result of Gaschütz: Corollary. G is p-solvable if and only if \(s(G,M)=\dim_ DH^ 1(G,M)\) for every simple kG-module M. These results have equivalent formulations in terms of modular representation theory.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 3995976
zbMATH Keywords: finite group
zbMATH Keywords: simple kG-module
zbMATH Keywords: split chief factor
zbMATH Keywords: chief series
zbMATH Keywords: p- solvable
author: Urs Stammbach
reviewed by: Chih-Han Sah
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90037-5
OpenAlex ID: W1990543394
cites work: Maximal subgroups of finite groups
cites work: Q5648522
cites work: Q3934684
cites work: Cohomological characterisations of finite solvable and nilpotent groups
cites work: On p-chief factors of finite groups
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description: scientific article; zbMATH DE number 3995976