On pro-\(p\) groups with potent filtrations.
type: Item
instance of: scholarly article
title: On pro-\(p\) groups with potent filtrations.
zbMATH Open document ID: 1182.20034
author: Primož Moravec
published in: Journal of Algebra
publication date: 2009-09-10T00:00:00Z
review text: \textit{G. A. Fernández-Alcober, J. González-Sánchez} and \textit{A. Jaikin-Zapirain} have recently defined [Isr. J. Math. 166, 393-412 (2008; Zbl 1153.20028)] the family of PF-groups. A potent filtration of the pro-\(p\) group \(G\) is a descending chain of closed subgroups \(N_i\), for \(i\in\mathbb{N}\), whose intersection is trivial, and such that \([N_i,G]\leq N_{i+1}\) and \([N_i,{_{p-1}G}]\leq N_{i+1}^p\) for all \(i\in\mathbb{N}\). Now \(G\) is said to be a PF-group if it has a potent filtration starting with \(N_0=G\). This generalizes both potent pro-\(p\) groups [see \textit{J. González-Sánchez} and \textit{A. Jaikin-Zapirain}, J. Algebra 276, No. 1, 193-209 (2004; Zbl 1056.20014)] and powerful pro-\(p\) groups [see \textit{A. Lubotzky} and \textit{A. Mann}, J. Algebra 105, 484-505 (1987; Zbl 0626.20010)]. The main result of the paper under review is a generalization of both a result of Lubotzky and Mann from the aforementioned paper, and a result of \textit{G. Ellis} [Trans. Am. Math. Soc. 353, No.~10, 4219-4234 (2001; Zbl 1032.20023)]. It is shown that if \(G\) is a PF-group of finite exponent, then the exponent of \(H_2(G,M)\) divides the exponent of \(G\) for every profinite trivial \(\|\widehat\mathbb{Z} G\|\)-module \(M\).
reviewed by: Andrea Caranti
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 5602652
zbMATH Keywords: Schur multipliers
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: PF-groups
zbMATH Keywords: potent filtrations
zbMATH Keywords: descending chains of closed subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.01.011
OpenAlex ID: W1983127867
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DOI: 10.1016/J.JALGEBRA.2009.01.011
description: scientific article; zbMATH DE number 5602652