Extending irreducible representations of normal subgroups
type: Item
instance of: scholarly article
title: Extending irreducible representations of normal subgroups
zbMATH Open document ID: 0613.20005
DOI: 10.1016/0021-8693(85)90203-0
author: Peter Schmid
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let H be a normal subgroup of a finite group G, F a field, and V an irreducible FH module. The author wishes to find necessary and sufficient conditions in order that V can be extended to an FG module. Let \(D=End_{FH}(V)^ 0\) be the opposite endomorphism ring of V and let Z denote the multiplicative group of the center of D. If the Schur index m(V) of V is relatively prime to the order of G/H, then the author associates to V a cohomology class \(w_ G(V)\in H^ 2(G/H,Z)\). The main theorem then states that V can be extended to an FG module if and only if \(w_ G(V)\) vanishes. The proof uses a Hochschild-Serre exact sequence and a ''Schur-Zassenhaus type'' splitting theorem. Several interesting facts are presented in the course of proving the theorem. If H is a Hall subgroup, this extension problem, without the assumption on the Schur index, had been considered by \textit{I. Isaacs} [J. Algebra 68, 54-74 (1981; Zbl 0449.20016)] and by \textit{E. Dade} [ibid. 72, 374-403 (1981; Zbl 0497.20001)].
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 3991752
zbMATH Keywords: irreducible FH module
zbMATH Keywords: endomorphism ring
zbMATH Keywords: Schur index
zbMATH Keywords: cohomology class
zbMATH Keywords: Hochschild-Serre exact sequence
zbMATH Keywords: splitting theorem
zbMATH Keywords: Hall subgroup
zbMATH Keywords: extension problem
reviewed by: E. Spiegel
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90203-0
OpenAlex ID: W2033113788
cites work: A structure theory for Jordan algebras
cites work: Isomorphisms of Clifford extensions
cites work: Character Values and Clifford Extensions for Finite Groups
cites work: Extending irreducible modules
cites work: Cohomology and Galois Theory. I. Normality of Algebras and Teichmuller's Cocycle
cites work: Some Properties of Characters of Finite Groups
cites work: Q3955560
cites work: An Extension Theorem for Characters
cites work: Automorphisms of Simple Algebras
cites work: Cohomology of Group Extensions
cites work: Endliche Gruppen I
cites work: Extensions of group representations over arbitrary fields
cites work: Twisted group algebras and Clifford extensions
cites work: Q4126536
description: scientific article; zbMATH DE number 3991752