Maximal subgroups of the minimal ideal of a free profinite monoid are free.
type: Item
instance of: scholarly article
title: Maximal subgroups of the minimal ideal of a free profinite monoid are free.
zbMATH Open document ID: 1220.20045
author: Benjamin Steinberg
published in: Israel Journal of Mathematics
publication date: 2010-06-29T00:00:00Z
review text: The author shows that the maximal subgroup of the minimal ideal of a finitely generated free profinite monoid is a free profinite group. This answers a question asked by S. W. Margolis in the 1990s. More generally, if \(\mathbf H\) is an extension-closed pseudovariety of finite groups that contains a cyclic \(p\)-group for infinitely many primes \(p\) and \(\overline{\mathbf H}\) is the pseudovariety of all finite monoids whose subgroups lie in \(\mathbf H\), then the maximal subgroup of the minimal ideal of a finitely generated free pro-\(\overline{\mathbf H}\) monoid is a free profinite group. Reviewer's remark. A further generalization has been obtained by \textit{A. Costa} and the author [Proc. Lond. Math. Soc. (3) 102, No. 2, 341-369 (2011; Zbl 1257.20054)].
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 20A15
zbMATH DE Number: 5728344
zbMATH Keywords: free profinite monoids
zbMATH Keywords: free profinite groups
zbMATH Keywords: minimal ideals
zbMATH Keywords: extension-closed pseudovarieties
zbMATH Keywords: pseudovarieties of finite monoids
MaRDI profile type: Publication
full work available at URL: s11856-010-0023-z
OpenAlex ID: W2009350950
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DOI: 10.1007/S11856-010-0023-Z
description: scientific article; zbMATH DE number 5728344