Profinite groups admitting just infinite quotients
type: Item
instance of: scholarly article
title: Profinite groups admitting just infinite quotients
zbMATH Open document ID: 1005.20025
published in: Monatshefte für Mathematik
publication date: 2002-06-10T00:00:00Z
review text: An infinite group is called just infinite if all of its proper quotients are finite. There has been much interest in just infinite groups in recent years, particularly in the field of pro-\(p\) groups where people like to think of just infinite pro-\(p\) groups as playing a role in pro-\(p\) group theory similar to the role played by finite simple groups in finite group theory. Indeed the German term for just infinite is `unendlich einfach' (infinitely simple). This notion is enhanced by the fact that every finitely generated pro-\(p\) group has a just infinite quotient. This is proved using Zorn's Lemma and the Frattini-series [Proposition II.5 in Linear pro-\(p\)-groups of finite width, Lect. Notes Math. 1674, Springer, \textit{G. Klaas, C. R. Leedham-Green} and \textit{W. Plesken} (1997; Zbl 0901.20013)]. In this paper the author addresses the question of which profinite groups admit just infinite quotients. He proves that any profinite group whose order (as a supernatural number) is divisible only by finitely many primes admits just infinite quotients. He also proves that the property of admitting a just infinite quotient is preserved when going to open subgroups and finite extensions. This is used to deduce that every virtually projective group (not necessarily finitely generated) admits a just infinite quotient. There are other results concerning profinite groups acting on trees. Instead of using the Frattini subgroup the proofs concentrate on the intersection of maximal normal subgroups.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 1753105
zbMATH Keywords: profinite groups
zbMATH Keywords: just infinite groups
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: just infinite quotients
zbMATH Keywords: open subgroups
zbMATH Keywords: finite extensions
zbMATH Keywords: virtually projective groups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: Frattini subgroup
author: Pavel Zalesskij
reviewed by: Rachel D. Camina
MaRDI profile type: Publication
full work available at URL: s006050200013
OpenAlex ID: W2023344825
DOI: 10.1007/S006050200013
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description: scientific article; zbMATH DE number 1753105