Central units in blocks and the odd \(Z_p^*\)-theorem.
type: Item
instance of: scholarly article
title: Central units in blocks and the odd \(Z_p^*\)-theorem.
zbMATH Open document ID: 1169.20007
author: Geoffrey R. Robinson
published in: Journal of Algebra
publication date: 2009-04-21T00:00:00Z
review text: Let \(p\) be an odd prime, \(G\) a finite group, \(x\) an element of order \(p\) in a \(p\)-Sylow subgroup \(P\). Glauberman's \(Z^*\) theorem states that if the element \(x\) commutes with none of its other conjugates, then \([x,G]\leq O^*_{p'}(G)\) holds. An analogue of this theorem for odd primes is that the subgroup \(O_{p'}(G)C_G(x)\) equals \(G\) if \(x\) is an isolated element in the subgroup \(P\), which follows from the classification of finite simple groups. In the present paper an independent proof is sought by means of representation theory. In the paper [J. Algebra 134, No. 2, 353-355 (1990; Zbl 0706.20012)] of the author it was shown that the property has a close relationship with units of prime order in the group ring. This idea is utilized also here: in the course of the investigations units of order \(p\) in the \(p\)-adic group ring are constructed valuable in proving the analogue of Glauberman's theorem and also of some independent interest.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 5545839
zbMATH Keywords: blocks
zbMATH Keywords: units of prime order
zbMATH Keywords: \(p\)-adic group rings
zbMATH Keywords: finite groups
zbMATH Keywords: Glauberman \(Z^*\)-theorem
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.10.017
OpenAlex ID: W1990028483
cites work: Characters of Relatively Projective Modules II
cites work: Brauer characters relative to a normal subgroup
cites work: Remarks on coherence and the Reynolds isometry
cites work: Q3027210
cites work: The \(Z_ p^*\)-theorem and units in blocks
DOI: 10.1016/J.JALGEBRA.2008.10.017
description: scientific article; zbMATH DE number 5545839