Perfect isometries for blocks with abelian defect groups and cyclic inertial quotients of order \(4\)
type: Item
instance of: scholarly article
title: Perfect isometries for blocks with abelian defect groups and cyclic inertial quotients of order \(4\)
zbMATH Open document ID: 0831.20010
author: Lluis Puig
author: Yoko Usami
published in: Journal of Algebra
publication date: 1996-02-08T00:00:00Z
review text: Let \(b\) be a block of a finite group \(G\) with abelian defect group \(P\). A conjecture by M. Broué says that \(b\) is isotypic to its Brauer correspondent \(c\) in \(N_G (P)\). The authors verify this conjecture in the special case where the inertial quotient \(N_G(P, e)/PC_G(P)\) is a cyclic group of order 4; here \(e\) denotes a block of \(C_G(P)\) such that \(e^G = b\). Combined with earlier results this implies that Broué's conjecture holds whenever \(|N_G (P,e)/PC_G(P)|\leq 4\). It follows that in these cases there exists a perfect isometry between \(b\) and \(c\); in particular, \(b\) and \(c\) have the same numbers of irreducible ordinary and modular characters, and all irreducible characters in \(b\) have height zero.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 763917
zbMATH Keywords: isotypy
zbMATH Keywords: Brauer's height zero conjecture
zbMATH Keywords: Alperin's weight conjecture
zbMATH Keywords: block with abelian defect group
zbMATH Keywords: finite groups
zbMATH Keywords: Brauer correspondent
zbMATH Keywords: inertial quotients
zbMATH Keywords: perfect isometry
zbMATH Keywords: modular characters
zbMATH Keywords: irreducible characters
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: jabr.1995.1059
OpenAlex ID: W2083151461
DOI: 10.1006/JABR.1995.1059
description: scientific article; zbMATH DE number 763917