On \(\omega\)-filtered vector spaces and their application to Abelian p- groups. I
type: Item
instance of: scholarly article
title: On \(\omega\)-filtered vector spaces and their application to Abelian p- groups. I
zbMATH Open document ID: 0573.20054
DOI: 10.1007/BF02567406
author: B. George
published in: Commentarii Mathematici Helvetici
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140006
review text: The socle G[p] of a p-group G is an \(\omega\)-filtered vector space with the filtration \((p^ nG)[p]\), \(n=0,1,2,... \). With applications to p- groups in mind the authors investigate \(\omega\)-filtered vector spaces over at most countable fields using some of the more sophisticated tools of set theory. It is not possible to explain here the numerous categorical concepts involved and the set theoretical concepts utilized. For the novice the carefully written paper forms an excellent introduction to certain set theoretical methods and filtered vector spaces. For the initiated we mention that an invariant \(\Gamma\) (X) of the \(\omega\)-filtered vector space X of uncountable regular dimension \(\kappa\) (mostly \(\kappa =\kappa_ 1)\) plays a central role. It is an element of the Boolean algebra of all subsets of \(\kappa\) modulo the ideal of thin subsets of \(\kappa\). It is obtained by writing X as the union of a smooth ascending chain of subspaces \(X_{\alpha}\) \((\alpha <\kappa)\) of X which are ''small'' i.e. \(\dim (X_{\alpha})<\kappa\). If E is the set of all \(\alpha\) such that \(X_{\alpha}\) is not a direct summand of every (filtered) subspace of X in which \(X_{\alpha}\) has codimension less than \(\kappa\) then \(\Gamma (X)=E\) modulo thin sets. The significance of the invariant \(\Gamma\) (X) may be seen from two results: \(\Gamma (X)=0\) if and only if X is a projective filtered vector space; \(\Gamma\) (X)\(\neq 1\) if and only if every countable subspace of X is contained in a closed countable subspace and X is Hausdorff in the topology induced by its filtration. However, the invariant \(\Gamma\) in no way pins down the space: there are a large number of non-isomorphic spaces all with the same \(\Gamma\) and the same basic subspace. Among the applications is the following: A p-group is weakly \(\omega_ 1\)- separable if it is separable and every countable subset is contained in a countable closed pure subgroup. Theorem: Let G be a separable p-grou…
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K45
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 3914587
zbMATH Keywords: socle
zbMATH Keywords: p-group
zbMATH Keywords: \(\omega \) -filtered vector spaces
zbMATH Keywords: invariant
zbMATH Keywords: uncountable regular dimension
zbMATH Keywords: Boolean algebra
zbMATH Keywords: thin subsets
zbMATH Keywords: smooth ascending chain of subspaces
zbMATH Keywords: projective filtered vector space
zbMATH Keywords: filtration
zbMATH Keywords: separable p- group
zbMATH Keywords: \(p^{\omega +1}\)-injective p-groups
zbMATH Keywords: \(p^{\omega +1}\)-projective groups
reviewed by: Adolf Mader
MaRDI profile type: Publication
OpenAlex ID: W2027123521
description: scientific article; zbMATH DE number 3914587