Density of indecomposable locally finite groups
type: Item
instance of: scholarly article
title: Density of indecomposable locally finite groups
DOI: 10.4171/RSMUP/68
author: S. Shelah
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2021-07-02T00:00:00Z
review text: Let \(\theta\) be a cardinal number. A group \(G\) is called \(\theta\)-indecomposable if, for every increasing sequence \(\langle G\mid i<\theta\rangle\) of subgroups with union \(G\), there is \(i <\theta\) such that \(G=G_i\). Moreover, a group \(E\) is said to be a locally finite existentially closed group if it contains an isomorphic copy of every finite group and any two isomorphic finite subgroups of \(E\) are conjugate in \(E\) (see the universal groups in [\textit{O. Kegel} and \textit{B. A. F. Wehrfritz}, Locally Finite Groups. New York: Elsevier (1973; Zbl 0259.20001)]). The main result of the paper is the following. Let \(G\) be a locally finite group. Then, under some very weak hypotheses on \(|G|\), there exists a locally finite existentially closed group \(H\) such that \(|H|=|G|\), \(G\leq H\) and such that it is \(\theta\)-indecomposable for any regular cardinal \(\theta\neq cf(|G|)\). We remark that this result is also used in two subsequent works by the same author.
reviewed by: Mattia Brescia
Mathematics Subject Classification ID: 20A10
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 7367420
zbMATH Keywords: model theory
zbMATH Keywords: applications of model theory
zbMATH Keywords: groups
zbMATH Keywords: locally finite groups
zbMATH Keywords: canonical extension
zbMATH Keywords: indecomposable group
MaRDI profile type: Publication
cites work: Strongly bounded groups of various cardinalities
cites work: Q5609457
cites work: Existentially closed structures in the power of the continuum
cites work: On universal locally finite groups
cites work: Complete Universal Locally Finite Groups
cites work: Q5675882
cites work: Existentially closed structures and Jensen's principle \(\diamond\)
cites work: Uncountable universal locally finite groups
cites work: A solution to the L space problem
cites work: Infinite Abelian groups, Whitehead problem and some constructions
cites work: A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals
cites work: Was Sierpinski right? I
cites work: Q4326402
cites work: More Jonsson algebras
cites work: Existentially closed locally finite groups
cites work: Universal structures
cites work: Compactness in singular cardinals revisited
cites work: Q3586891
cites work: Q4428699
cites work: The cofinality spectrum of the infinite symmetric group
cites work: Algebraically closed groups of large cardinality
cites work: Logic colloquium '84. Proceedings of the Colloquium held in Manchester, U.K., July 1984
cites work: Q3706144
cites work: Partitioning pairs of countable ordinals
description: scientific article; zbMATH DE number 7367420