Ein Untergruppensatz für modulare Gruppen
type: Item
instance of: scholarly article
title: Ein Untergruppensatz für modulare Gruppen
zbMATH Open document ID: 0483.20015
DOI: 10.1007/BF01301933
author: Ragmar Rudolph
published in: Monatshefte für Mathematik
publication date: 1982-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178091
review text: Recently A. Yu. Ol'shanskiĭproved the existence of so-called ''Tarski-monsters'', i.e. infinite groups with every proper subgroup of prime order [Math. USSR, Izv. 16, 279-289 (1981), translation from Izv. Akad. Nauk SSR, Ser. Mat. 44, 309--321 (1980; Zbl 0475.20025)]. A group \(G\) is called modular if the lattice of subgroups, \(L(G)\), of \(G\) is modular. In the paper under review we prove that every Infinite periodic 2-generator modular group \(G\) with generators \(a\) and \(b\), such that the cyclic subgroups generated by \(a\) and \(b\) intersect trivially, contains a Tarski-monster as a subgroup. An immediate consequence of the above is the fact that such a group is the direct sum of finitely many Tarski-monsters and a finite modular group; furthermore the orders of elements in different direct summands are relatively prime. Thus by a well-known theorem of Suzuki \(L(G)\) has a corresponding decomposition.
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 3758578
zbMATH Keywords: finitely generated
zbMATH Keywords: infinite periodic modular groups
zbMATH Keywords: Tarski-monsters
zbMATH Keywords: infinite groups with every proper subgroup of prime order
zbMATH Keywords: lattice of subgroups
zbMATH Keywords: 2-generator modular group
zbMATH Keywords: direct sum of Tarski-monsters
MaRDI profile type: Publication
cites work: Finiteness properties of groups
cites work: Q5840059
cites work: Q3208861
cites work: AN INFINITE GROUP WITH SUBGROUPS OF PRIME ORDERS
cites work: Q3867969
cites work: Die Endlichkeit einiger modularer Gruppen
cites work: On the Lattice of Subgroups of Finite Groups
cites work: Sul comportamente degli elementi periodici in un gruppo di Dedekind infinito
OpenAlex ID: W137545056
description: scientific article; zbMATH DE number 3758578