Uncountable groups have many nonconjugate subgroups
type: Item
instance of: scholarly article
title: Uncountable groups have many nonconjugate subgroups
zbMATH Open document ID: 0629.20001
DOI: 10.1016/0168-0072(87)90016-9
author: S. Shelah
published in: Annals of Pure and Applied Logic
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/26902
review text: The whole of this substantial paper is devoted to the proof of the theorem: Main theorem. If G is a group of cardinality \(\lambda\), \(\lambda\) an uncountable cardinal, and \(\mu =Min\{\mu:\) \(2^{\mu}\geq \lambda \}\), then \(nc_{\leq \mu}(G)\geq \lambda\). (Where \(nc_{\leq \mu}(G)\) is the number of pairwise nonconjugate subgroups of G of power \(\mu\).) This theorem has the interesting conclusion: Conclusion. If \(\lambda\) is an uncountable cardinal and G a group of cardinality \(\lambda\) then G has at least \(\lambda\) pairwise nonconjugate subgroups of power less than \(\lambda\). This paper completes the work begun by the author in a previous paper [Algebra Univers. 16, 131-146 (1983; Zbl 0521.20015)] where the result was proved under GCH and for many values of \(\lambda\) (for every G). The techniques used are those of mathematical logic, and the author thoughtfully provides an appendix for nonlogicians containing the facts from mathematical logic that are required for an understanding of the paper.
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 4023544
zbMATH Keywords: uncountable cardinal
zbMATH Keywords: number of pairwise nonconjugate subgroups
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
OpenAlex ID: W2150865059
cites work: Model theory
cites work: A weak version of \(\lozenge\) which follows from \(2^{\aleph_0}<2^{\aleph_1}\)
cites work: Some theorems of set theory and their topological consequences
cites work: On the number of non conjugate subgroups
cites work: Word problems II. The Oxford book
cites work: Classification theory and the number of non-isomorphic models
cites work: Proper forcing
description: scientific article; zbMATH DE number 4023544