Normal subgroups and heights of characters
type: Item
instance of: scholarly article
title: Normal subgroups and heights of characters
zbMATH Open document ID: 0867.20011
author: Masafumi Murai
published in: Journal of Mathematics of Kyoto University
publication date: 1997-08-10T00:00:00Z
review text: This paper deals with relations between the blocks of a finite group \(G\) and those of a normal subgroup \(N\) resp. of the factor group \(\overline G=G/N\). The first part of the paper is devoted to the notation ``domination'', which means: Given an indecomposable \(OG\)-module \(V\) (\(O\) a large field of characteristic \(p>0\) or the ring of integers in a \(p\)-modular system for \(G\)) which is indecomposable as an \(ON\)-module, then a (\(p\)-)block \(B\) of \(G\) dominates a block \(\overline{B}\) of \(\overline G\) through \(V\), if there exists an \(O\overline G\)-module \(X\) in \(\overline B\) such that \(V\otimes\text{infl}(X)\) belongs to \(B\). (\(\text{infl}(X)\) means the inflation of \(X\) to \(G\).) This is equivalent, as the author shows, to the fact that \(B\) covers the block \(b\) of \(ON\) to which \(V_N\) belongs. Furthermore each block \(\overline B\) of \(\overline G\) is dominated through \(V\) by a unique block \(B\) of \(G\) and has a defect group contained in \(DN/N\) where \(D\) is a defect group of \(B\). In the latter statement equality holds for some block \(\overline B\) which is dominated by \(B\) through \(V\). Applications are given when \(V\) affords an irreducible character (ordinary or Brauer) which remains irreducible as a character of \(N\). In the second part of the paper which relies on some of the investigations in the first part, the author investigates heights of characters. He mainly proves: Let \(\xi=\text{Irr}(N)\) belong to a block \(b\) of \(N\) and let \(B\) be a block of \(G\) with defect group \(D\) covering \(b\). Furthermore, let \(\widehat{b}\) be the unique block of \(DN\) covering \(b\) and let \(T_G(b)\) be the inertial group of \(b\) in \(G\). Then the following three numbers are equal: (i) \(\min\{\text{ht}(\chi)-\text{ht}(\xi)\mid\chi\in\text{Irr}(B)\) lying over \(\xi\}\), (ii) \(\min\{\text{ht}(\zeta)-\text{ht}(\xi)\mid\zeta\in\text{Irr}(\widehat b)\) lying over a \(T_G(b)\)-conjugate of \(\xi\}\), …
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 950577
zbMATH Keywords: blocks of finite groups
zbMATH Keywords: domination
zbMATH Keywords: indecomposable modules
zbMATH Keywords: \(p\)-modular systems
zbMATH Keywords: inflation
zbMATH Keywords: irreducible characters
zbMATH Keywords: heights of characters
zbMATH Keywords: inertial groups
zbMATH Keywords: Brauer characters
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518602
OpenAlex ID: W1566299247
DOI: 10.1215/KJM/1250518602
description: scientific article; zbMATH DE number 950577