The hypercenter of a profinite group.
type: Item
instance of: scholarly article
title: The hypercenter of a profinite group.
zbMATH Open document ID: 1307.20026
published in: Beiträge zur Algebra und Geometrie
publication date: 2014-10-30T00:00:00Z
review text: The hypercenter of a group \(G\) is usually defined as the union of the members of the upper central series. However, there are pronilpotent, non-trivial groups with trivial center. For a profinite group \(G\), it thus makes sense to define the (upper) hypercenter \(Z^\infty(G)\) as the set of elements that map into the hypercenter of the (finite) quotients modulo all open normal subgroups. The paper under review gives various characterizations of \(Z^\infty(G)\), as the largest subgroup normalizing every Sylow subgroup; as the largest pronilpotent normal subgroup \(Z\) such that \(G\) induces on each Sylow \(p\)-subgroup of \(Z\) a pro-\(p\) group of automorphisms, for all primes \(p\); as the set of elements \(x\in G\) such that \(\lim_{n\to\infty}[x,\underbrace{g,\ldots,g}_{n}]=1\) for every \(g\in G\). Moreover, a closed, core-free subgroup of \(G\) is shown to be contained in \(Z^\infty(G)\) if and only if it permutes with each Sylow subgroup and each Sylow normalizer.
reviewed by: Andrea Caranti
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 6362522
zbMATH Keywords: profinite groups
zbMATH Keywords: hypercenter
zbMATH Keywords: Sylow theory
zbMATH Keywords: Engel elements
zbMATH Keywords: permutable subgroups
MaRDI profile type: Publication
full work available at URL: s13366-013-0149-3
OpenAlex ID: W2001326943
cites work: Engelsche Elemente Noetherscher Gruppen
cites work: Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields
cites work: Endliche Gruppen I
cites work: The embedding of quasinormal subgroups in finite groups
cites work: Q4490702
cites work: Subgroups permutable with all Sylow subgroups
cites work: Q4349107
DOI: 10.1007/S13366-013-0149-3
author: Peter Schmid
description: scientific article; zbMATH DE number 6362522