Homogeneity of infinite permutation groups
type: Item
instance of: scholarly article
title: Homogeneity of infinite permutation groups
zbMATH Open document ID: 0669.20002
DOI: 10.1007/BF01633987
author: S. Shelah
author: Simon Thomas
published in: Archive for Mathematical Logic
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Omega\) be an infinite set, \(\lambda <| \Omega |\) an arbitrary cardinal, and G a group of permutations of \(\Omega\). G is said to be \(\lambda\)-homogeneous if for each pair A,B\(\subseteq \Omega\) with \(| A| =| B| =\lambda\), there exists \(g\in G\) with \(A^ g=B\). It is known that if G is m-homogeneous for some \(m\in \omega\), then G is also n-homogeneous for all \(n<m\). Recently, \textit{P. M. Neumann} [Bull. Lond. Math. Soc. 20, 305-312 (1988; Zbl 0644.20007)] proved that if G is \(\lambda\)-homogeneous for some \(\lambda\geq \omega\), then G is n- homogeneous (even n-transitive) for all \(n\in \omega\). Using Martin's Axiom (MA) and combinatorial arguments, the authors show: Theorem. Assume (MA) and \(\omega_ 2<2^{\omega}\). Let \(\omega_ 1<\kappa <2^{\omega}\). Then there exists a group G of permutations of \(\kappa\) such that G is \(\lambda\)-homogeneous for all \(\omega <\lambda <\kappa\), but not \(\omega\)-homogeneous. - This answers a question of P. M. Neumann [loc. cit.] in the negative. The authors remark that it remains open whether it is consistent with (ZFC) that every \(\omega_ 1\)-homogeneous permutation group is \(\omega\)-homogeneous.
reviewed by: Manfred Droste
Mathematics Subject Classification ID: 20B22
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 20B07
zbMATH DE Number: 4095736
zbMATH Keywords: homogeneity
zbMATH Keywords: group of permutations
zbMATH Keywords: n-homogeneous
zbMATH Keywords: n-transitive
zbMATH Keywords: Martin's Axiom
zbMATH Keywords: \(\omega _ 1\)-homogeneous permutation group
MaRDI profile type: Publication
cites work: Weak n-Homogeneity Implies Weak (n - 1)-Homogeneity
cites work: Homogeneity of Infinite Permutation Groups
cites work: Endliche \(k\)-homogene Permutationsgruppen
full work available at URL: https://doi.org/10.1007/bf01633987
OpenAlex ID: W2014426589
Recommended article: Homogeneity of Infinite Permutation Groups
Recommended article: A Remark on the Homogeneity of infinite Permutation Groups
Recommended article: ON -HOMOGENEOUS, BUT NOT -TRANSITIVE PERMUTATION GROUPS
Recommended article: Homogeneity in infinite permutation groups
Recommended article: Q3033949
Recommended article: Seperation Theorems for infinite Permutation Groups
Recommended article: Q4226363
Recommended article: Q3726327
Recommended article: Groups of order-automorphisms of the rationals with prescribed scale type
Recommended article: On the representation of a group of finite order as a permutation group, and on the composition of permutation groups.
description: scientific article; zbMATH DE number 4095736