The \(Z_ p^*\)-theorem and units in blocks
type: Item
instance of: scholarly article
title: The \(Z_ p^*\)-theorem and units in blocks
zbMATH Open document ID: 0706.20012
DOI: 10.1016/0021-8693(90)90058-V
author: Geoffrey R. Robinson
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: The paper under review points out a surprising and interesting connection between the \(Z^*_ p\)-theorem in local group theory and the isomorphism problem for group rings. Let \(p\) be a prime, let \(G\) be a finite group, let \(x\) be an element of order \(p\) in \(G\), and suppose that \(x^ g=x\) for any \(g\in G\) such that \(x^ g\) and \(x\) lie in a common Sylow \(p\)-subgroup. The \(Z^*_ p\)-conjecture says that \(G=O_{p'}(G)C_ G(x)\) in this case. It was proved for \(p=2\) by G. Glauberman and is known to be a consequence of the classification of finite simple groups for arbitrary \(p\). Without using the classification, the author constructs, for a minimal counterexample \(G\) and a prime \(p\geq 5\), a unit \(u\) of order \(p\) and augmentation 1 in the center of the principal block \(B_ 0(G)\) of \(G\) over \({\mathbb{Z}}_{(p)}\) such that \(u\) is not conjugate to \(e_ 0g\) for any \(g\in G\) (where \(e_ 0\) denotes the principal block idempotent). The existence of such a unit contradicts the following conjecture by K. Roggenkamp and L. Scott: Let \(G_ 1\), \(G_ 2\) be finite groups such that \(B_ 0(G_ 1)=B_ 0(G_ 2)\) as augmented algebras, and let \(D_ 1\), \(D_ 2\) be Sylow \(p\)-subgroups of \(G_ 1\), \(G_ 2\), respectively. Then \(e_ 0D_ 1\) and \(e_ 0D_ 2\) are conjugate via a unit in \(B_ 0(G_ 1)\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 4158933
zbMATH Keywords: normalized units
zbMATH Keywords: defect groups
zbMATH Keywords: \(Z^ *_ p\)-theorem
zbMATH Keywords: isomorphism problem for group rings
zbMATH Keywords: finite groups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: augmentation 1
zbMATH Keywords: principal block
zbMATH Keywords: principal block idempotents
zbMATH Keywords: augmented algebras
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
cites work: Central elements in core-free groups
cites work: Remarks on coherence and the Reynolds isometry
cites work: Q4141384
full work available at URL: 0021-8693(90)90058-v
OpenAlex ID: W2079436418
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description: scientific article; zbMATH DE number 4158933