A remark on covering blocks
type: Item
instance of: scholarly article
title: A remark on covering blocks
zbMATH Open document ID: 0596.20010
DOI: 10.1016/0021-8693(86)90179-1
author: Reinhard Knörr
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Generalizing well known results of E. C. Dade and I. M. Isaacs the author shows in the main theorem of this note: Let N be a normal subgroup of G such that G/N is a p-group, and let b be a p-block of N. Then there is precisely one p-block B of G covering b. If a defect group D of B is abelian, then the following statements are equivalent: (1) There is a G- stable module in b, (2) b is G-stable, (3) \(G=DN\), (4) If \(\chi\in B\) is an irreducible ordinary or Brauer character of G, then \(\chi |_ N\) is irreducible, (5) Every ordinary irreducible character \(\psi\) of b has precisely \(q=| G:N|\) different extensions to G, all belonging to B. Every irreducible Brauer character \(\rho\) of b has precisely one extension \({\hat \rho}\) to G and \({\hat \rho}\in B.\) Furthermore, if these equivalent conditions are satisfied, then the decomposition matrix of B is obtained from the decomposition matrix of b by repeating each row q times, and for the Cartan matrices, one has \(C_ B=q^ Cb\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3959746
zbMATH Keywords: extendibility of irreducible characters
zbMATH Keywords: normal subgroup
zbMATH Keywords: p-group
zbMATH Keywords: p- block
zbMATH Keywords: defect group
zbMATH Keywords: G-stable module
zbMATH Keywords: ordinary irreducible character
zbMATH Keywords: irreducible Brauer character
zbMATH Keywords: decomposition matrix
zbMATH Keywords: Cartan matrices
reviewed by: Gerhard O. Michler
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90179-1
OpenAlex ID: W2047032034
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cites work: On the vertices of irreducible modules
description: scientific article; zbMATH DE number 3959746