On linear just infinite pro-\(p\) groups
type: Item
instance of: scholarly article
title: On linear just infinite pro-\(p\) groups
zbMATH Open document ID: 1018.20022
DOI: 10.1016/S0021-8693(02)00024-8
author: Andrei Jaikin-Zapirain
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The author studies linearity of pro-\(p\) groups over commutative rings and over commutative profinite rings, in particular linearity of just infinite pro-\(p\) groups, i.e. groups whose proper quotients are finite. A group \(G\) is called hereditarily just infinite if every proper quotient of any subgroup of finite index of \(G\) is finite. Among others the following results are proved. Proposition 1.1. Let \(G\) be a just infinite pro-\(p\) group. Then \(G\) is insoluble if and only if every non-trivial normal (not necessarely closed) subgroup of \(G\) is open. Corollary 1.2. Let \(G\) be an insoluble hereditarily just infinite pro-\(p\) group. Then \(G\) is hereditarily just infinite as an abstract group. Corollary 1.5. Let \(G\) be a linear just infinite pro-\(p\) group over some commutative profinite ring. Then \(G\) is linear either over the \(p\)-adic integers \(\mathbb{Z}_p\) or over the power series ring \(\mathbb{F}_pt\), where \(\mathbb{F}_p\) is the field of \(p\) elements. Theorem 1.8. Let \(G\) be a \(p\)-adic analytic pro-\(p\) group. Then \(G\) is linear over a field of positive characteristic if and only if \(G\) is virtually Abelian.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20G25
zbMATH DE Number: 1868012
zbMATH Keywords: just infinite pro-\(p\) groups
zbMATH Keywords: linear pro-\(p\) groups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: hereditarily just infinite groups
reviewed by: Pavel Zalesskij
MaRDI profile type: Publication
cites work: Q4264394
cites work: On some \(\Lambda\)-analytic pro-\(p\) groups
cites work: Non-Abelian free pro-\(p\)-groups cannot be represented by 2-by-2 matrices
cites work: A non-abelian free pro-\(p\) group is not linear over a local field
cites work: Normal subgroups in substitution groups of formal power series
cites work: Linear pro-\(p\)-groups of finite width
cites work: Compact subgroups of linear algebraic groups
cites work: New horizons in pro-\(p\) groups
cites work: Q4385471
cites work: Q4217279
cites work: Q3786586
cites work: Q3997989
cites work: A.D. Alexandrov spaces with curvature bounded below
cites work: Q5676405
cites work: Q3961610
cites work: Q5603302
full work available at URL: s0021-8693(02)00024-8
OpenAlex ID: W2038072645
description: scientific article; zbMATH DE number 1868012