Normal subgroups in substitution groups of formal power series
type: Item
instance of: scholarly article
title: Normal subgroups in substitution groups of formal power series
zbMATH Open document ID: 0964.20016
author: Benjamin Klopsch
published in: Journal of Algebra
publication date: 2001-04-19T00:00:00Z
review text: Let \(R\) be a commutative ring with 1. This paper is concerned with the group \(N(R):=t+t^2Rt\) of normalized formal power series over \(R\) under substitution. If \(K\) is a finite field, \(N(K)\) is known as a Nottingham group that possesses a list of peculiar properties. The author studies the normal structure of \(N(R)\) in general for a commutative ring \(R\) with unity. If \(2\) is invertible in \(R\), the author proves that every normal subgroup of \(N(R)\) is closed. He gives even more precise information on the normal structure that looks especially nice if \(R=K\) is a field of characteristic \(p>2\). Put \(N_k(R):=t+t^{k+1}Rt\), \(k\in\mathbb{N}\). Then the following theorem is proved. Theorem 1.3. Let \(K\) be a field of characteristic \(p\neq 2\). Then every (abstract) closed normal subgroup of \(N(K)\) is closed. Moreover, for every non-trivial normal subgroup \(H\triangleleft N(K)\) there exists \(k\in\mathbb{N}\) such that one of the following holds (i) \(N_{k+1}(K)\subseteq HN_k(K)\) where \(k\not\equiv 1\pmod p\); (ii) \(N_{k+2}(K)\subseteq HN_k(K)\) where \(k\equiv 1\pmod p\). Further, a type of the congruence subgroup problem is studied, i.e., the relation between the groups \(\widehat{N(R)}\) and \(N(\widehat R)\). First the author shows that a subgroup of finite index of \(N(R)\) is open in the topology given by \(N_k(R)\). Using this the author deduces that \(\widehat{N(R)}=N(\widehat R)\) for a saturated ring \(R\), i.e., a ring \(R\) in which every Abelian subgroup of finite index contains an ideal of finite index. Finally, it is shown that for \(R=\mathbb{F}_pt\) the groups \(\widehat{N(R)}\) and \(N(\widehat R)\) are not isomorphic.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E15
zbMATH DE Number: 1475239
zbMATH Keywords: profinite groups
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: Nottingham group
zbMATH Keywords: normal subgroups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: congruence subgroups
zbMATH Keywords: formal power series
reviewed by: Pavel Zalesskij
MaRDI profile type: Publication
full work available at URL: jabr.1999.8219
OpenAlex ID: W2023799861
cites work: Subgroups of the Nottingham group
cites work: Substitution Groups of Formal Power Series
cites work: Q3818488
cites work: Linear pro-\(p\)-groups of finite width
cites work: Q4891344
cites work: Q4217279
DOI: 10.1006/JABR.1999.8219
description: scientific article; zbMATH DE number 1475239