Pro-\(p\) groups with few normal subgroups.
type: Item
instance of: scholarly article
title: Pro-\(p\) groups with few normal subgroups.
zbMATH Open document ID: 1206.20030
author: J. Martínez
published in: Journal of Algebra
publication date: 2009-04-21T00:00:00Z
review text: The authors study pro-\(p\) groups with few normal subgroups in the paper. There are several notions regarding this and the authors study their interrelation as well as connections with existent notions of finite coclass, just infinity, obliquity. A pro-\(p\) group \(G\) has finite coclass if the index of the \(i\)-th term of the lower central series \([G:\gamma_i(G)]\leq p^{i+c-1}\) for some constant \(c\). Put \(\mu_i(G)=\gamma_i(G)\cap\bigcap_{N\triangleleft G,\;N\not\leq\gamma_i(G)}N\). A pro-\(p\) group \(G\) has finite obliquity if \(\sup_i[\gamma_i(G):\mu_i(G)]\) is finite. The authors consider the following conditions on a pair of open normal subgroups \(M\) and \(N\) of a pro-\(p\) group \(G\): (a) There exists a constant \(c\) such that for all \(N,M\) with \(M\not\subset N\) one has \(|N:M\cap N|\leq p^c\). (b) There exists \(c\) such that \(|N:M\cap N|\leq p^c\) for all pairs of subgroups \(M,N\) with the same index in \(G\). (c) For every \(N\) there is a constant \(c\) depending on \(N\) such that \(|N:M\cap N|\leq p^c\) if \(M\not\subset N\). A pro-\(p\) group of finite coclass satisfies the condition (b). The authors prove the following results: 1. Condition (a) is equivalent to \(G\) having finite obliquity. 2. Condition (b) is equivalent to the condition of Constant Normal Subgroup Growth, i.e., to the existence of a constant \(l\) such that \(G\) has at most \(l\) normal subgroups of any given index. 3. Condition (c) is equivalent to \(G\) being just infinite. The second part of the paper introduces the notion of a periodic map on the lattice of subgroups of \(G\). A periodic map on a pro-\(p\) group \(G\) is a map \(\tau\colon M\to G\) from an open normal subgroup \(M\) of \(G\) such that \(\tau(M)\) is an open subgroup of \(G\) and for every open normal subgroup \(H\) of \(G\) contained in \(\tau(M)\), the preimage \(\tau^{-1}(N)\) is an open normal subgroup of \(G\) with \(|G:\tau^{-1}(N)|<|G:N|\). The authors prove that the existence of a peri…
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 5545842
zbMATH Keywords: just infinite pro-\(p\) groups
zbMATH Keywords: normal subgroup growth
zbMATH Keywords: open normal subgroups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: groups of finite coclass
zbMATH Keywords: groups of finite width
zbMATH Keywords: groups of finite obliquity
zbMATH Keywords: periodic maps
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: branch groups
reviewed by: Pavel Zalesskij
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.10.012
OpenAlex ID: W2133116462
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DOI: 10.1016/J.JALGEBRA.2008.10.012
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description: scientific article; zbMATH DE number 5545842