On prosolvable subgroups of profinite free products and some applications
type: Item
instance of: scholarly article
title: On prosolvable subgroups of profinite free products and some applications
zbMATH Open document ID: 0827.20038
DOI: 10.1007/BF02567982
author: Florian Pop
published in: Manuscripta Mathematica
publication date: 1995-03-30T00:00:00Z
full work available at URL: https://eudml.org/doc/156042
review text: Let \(G\) be a profinite group, and \(\mathfrak G\) a family of closed subgroups of \(G\) closed under the conjugate action of \(G\). A finite \({\mathfrak G}\)- embedding problem for \(G\) is a diagram \(EP_{\mathfrak G}=(\gamma:G\to A,\alpha:B\to A,{\mathfrak B},\{\beta_\Gamma\}_{\Gamma\in{\mathfrak G}})\), where \(A\), \(B\) are finite groups, \(\gamma\), \(\alpha\) are surjective homomorphisms, and \(\mathfrak B\) is a conjugacy-closed family of subgroups of \(B\) such that every \(\Gamma\in{\mathfrak G}\) has a homomorphism into a member of \(\mathfrak B\) lifting \(\gamma\). If there exists \(\beta:G\to B\) with \(\gamma=\alpha\beta\), then \(\beta\) is called a solution of \(EP_{\mathfrak G}\), and if moreover \(\beta(\mathfrak G)\subset{\mathfrak B}\) holds, \(\beta\) is called a strong solution. We say that \(G\) is [strongly] \(\mathfrak G\)- projective if every finite \(\mathfrak G\)-embedding problem for \(G\) has a [strong] solution. It is shown that, under some technical conditions, any not-too-small prosolvable (or any finite) subgroup of a strongly \(\mathfrak G\)-projective group \(G\) must be contained in some member of \(\mathfrak G\) (Theorem 2). The author also applies this result to show that a certain general hypothesis on \(\mathfrak G\) is sufficient for a \(\mathfrak G\)-projective group to be strongly \(\mathfrak G\)-projective (Theorem 3). This paper provides a step to another article of the author: ``Embedding problems over large fields'' [Ann. Math., II. Ser. (to appear)].
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20J15
zbMATH DE Number: 724680
zbMATH Keywords: profinite free products
zbMATH Keywords: relatively projective groups
zbMATH Keywords: prosolvable subgroups
zbMATH Keywords: strongly projective groups
zbMATH Keywords: embedding problems
zbMATH Keywords: profinite groups
reviewed by: Hiroaki Nakamura
MaRDI profile type: Publication
cites work: A subgroup theorem for free products of pro-finite groups
cites work: On Closed Subgroups of Free Products of Profinite Groups
cites work: Q3726225
cites work: The absolute Galois group of a pseudo p-adically closed field.
cites work: Torsion elements and centralizers in free products of profinite groups.
cites work: Endliche Gruppen I
cites work: Prosolvable subgroups of free products of profinite groups
cites work: Über eine algebraische Kennzeichnung der Henselkörper.
cites work: On Grothendieck's conjecture of birational anabelian geometry
Recommended article: Q4695835
Recommended article: Projective products of profinite groups
Recommended article: On small profinite groups
Recommended article: Subgroups of a Direct Product of Groups that Satisfy the Strong Frattini Argument
Recommended article: Q3724559
Recommended article: Closed subgroups of free profinite monoids are projective profinite groups
Recommended article: THE PROFINITE COMPLETION OF A PROFINITE PROJECTIVE GROUP
Recommended article: K-projectivity in nonabelian groups
Recommended article: Q3710679
Recommended article: An axiomatizable profinite group with infinitely many open subgroups of index 2
description: scientific article; zbMATH DE number 724680