Fusion in profinite groups
type: Item
instance of: scholarly article
title: Fusion in profinite groups
zbMATH Open document ID: 0960.20015
DOI: 10.1007/BF02505917
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2001-02-13T00:00:00Z
review text: The first aim of the paper under review is to establish the basic theory of fusion and transfer for profinite groups, extending and generalizing the classical theory for finite groups. (For the latter, see for instance Sections~2 and 3 of the book by \textit{D. Gorenstein, R. Lyons} and \textit{R. Solomon} [The classification of the finite simple groups. No. 2. Part I, Chapter G: General group theory, (Mathematical Surveys and Monographs. Vol. 40, No. 2, Am. Math. Soc., Providence, RI) (1996; Zbl 0836.20011)].) The following question is then considered. Suppose \(\mathcal P\) is a class of pro-\(p\) groups, for some prime \(p\). What properties of \(\mathcal P\) imply that if the group \(S\in\mathcal P\) is isomorphic to a Sylow \(p\)-subgroup (or a weakly/strongly closed subgroup thereof) of the profinite group \(G\), then it controls transfer or fusion in \(G\)? An interesting result in this direction is the following. If the Sylow \(p\)-subgroup \(S\) of the profinite group \(G\) is powerful, then \(S\) controls transfer in \(G\). For the theory of powerful (pro-)\(p\)-groups, see [\textit{J. D. Dixon, M. P. F. du Sautoy, A. Mann} and \textit{D. Segal}, Analytic pro-\(p\) groups. 2nd ed., (Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, Cambridge) (1999; Zbl 0934.20001)].
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 1472058
zbMATH Keywords: profinite groups
zbMATH Keywords: fusion
zbMATH Keywords: transfer
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: powerful \(p\)-groups
author: Anna Luisa Gilotti
author: Luis Ribes
author: Luigi Serena
reviewed by: Andrea Caranti
MaRDI profile type: Publication
cites work: Q3997907
cites work: Strongly Closed Subgroups and Strong Fusion
cites work: Q5577154
cites work: Endliche Gruppen I
cites work: Q5523021
cites work: Q4856490
cites work: Q3712472
cites work: Character-theoretic transfer
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description: scientific article; zbMATH DE number 1472058