Pro-\(p\) groups of rank 3 and the question of Iwasawa.
type: Item
instance of: scholarly article
title: Pro-\(p\) groups of rank 3 and the question of Iwasawa.
zbMATH Open document ID: 1194.20031
author: Ilir Snopce
published in: Archiv der Mathematik
publication date: 2009-05-06T00:00:00Z
review text: Let \(p\) be a prime, and for any finitely generated pro-\(p\) group \(G\) let \(d(G)\) denote the minimal number of (topological) generators for \(G\). The aim of this short paper is to solve a special instance of the following classification problem: determine up to isomorphism all finitely generated pro-\(p\) groups \(G\) with the property that \(d(H)=d(G)\) for all open subgroups \(H\) of \(G\). This problem arises naturally from a more general question, attributed to Iwasawa; cf.\ \textit{D.~Dummit} and \textit{J.~Labute} [Invent. Math. 73, 413-418 (1983; Zbl 0546.20021)]. For given \(n\in\mathbb{N}\), what are the finitely generated pro-\(p\) groups \(G\) such that \(d(H)-n=|G:H|(d(G)-n)\) for every open subgroup \(H\) of \(G\)? Indeed, if one restricts attention to the category of \(p\)-adic analytic pro-\(p\) groups, one deals precisely with the pro-\(p\) groups \(G\) such that \(d(H)=n\) is constant as \(H\) runs through all open subgroups of \(G\). The main result of the paper under review is that, for \(p>3\), the pro-\(p\) groups \(G\) with \(d(H)=3\) for all open subgroups \(H\) of \(G\) are up to isomorphism (i) the free Abelian pro-\(p\) group \(G_0=\mathbb{Z}_p^3\) of rank \(3\) and (ii) the \(3\)-dimensional metabelian groups \(G_1(s)\), \(s\in\mathbb{N}\), admitting presentations \[ \langle x,y_1,y_2\mid [y_1,y_2]=1,\;[y_1,x]=y_1^{p^s},\;[y_2,x]=y_2^{p^s}\rangle. \] The proof given can be sketched as follows. The restriction \(p>3\) allows one to deduce that any pro-\(p\) group of the desired type is torsion-free. The author then makes use of a result of \textit{A. Lubotzky} and \textit{A. Mann} [J. Algebra 105, 506-515 (1987; Zbl 0626.20022)] which shows that groups of the desired type are soluble. Finally, he uses an effective classification of \(3\)-dimensional soluble torsion-free pro-\(p\) groups for \(p>3\), which was given by \textit{J.\ González-Sánchez} and \textit{B.\ Klopsch} [J. Group Theory 12, No. 5, 711-734 (2009; Zbl 1183.20030)].
reviewed by: Benjamin Klopsch
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 5551222
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: numbers of generators
MaRDI profile type: Publication
full work available at URL: s00013-008-2964-6
OpenAlex ID: W2033067564
DOI: 10.1007/S00013-008-2964-6
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description: scientific article; zbMATH DE number 5551222