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Abelian GroupsAlgebraically Compact Groups

Abelian Groups: Algebraically Compact Groups [In the preceding chapter we have encountered groups that were summands in every group containing them as pure subgroups: the pure-injective groups. In this chapter, we collect a large amount of additional information about these groups. Interestingly, these are precisely the summands of groups admitting a compact group topology, and the reduced ones are nothing else than the groups complete in the ℤ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}$$ \end{document}-adic topology. From Sect. 4 in Chapter 5 we know that every group can be embedded as a pure subgroup in a pure-injective (i.e., in an algebraically compact) group, and here we show that the significance of this embedding is enhanced by the fact that minimal embeddings exist and are unique up to isomorphism. Thus the theory of algebraically compact groups runs, in many respects, parallel to the theory of injective groups, a fact that was first pointed out by Maranda [1].] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Abelian GroupsAlgebraically Compact Groups

Part of the Springer Monographs in Mathematics Book Series
Fuchs, László
Springer Journals — Jun 12, 2015
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29 pages

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Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2015
ISBN
978-3-319-19421-9
Pages
183 –212
DOI
10.1007/978-3-319-19422-6_6
Publisher site
See Chapter on Publisher Site

Abstract

[In the preceding chapter we have encountered groups that were summands in every group containing them as pure subgroups: the pure-injective groups. In this chapter, we collect a large amount of additional information about these groups. Interestingly, these are precisely the summands of groups admitting a compact group topology, and the reduced ones are nothing else than the groups complete in the ℤ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}$$ \end{document}-adic topology. From Sect. 4 in Chapter 5 we know that every group can be embedded as a pure subgroup in a pure-injective (i.e., in an algebraically compact) group, and here we show that the significance of this embedding is enhanced by the fact that minimal embeddings exist and are unique up to isomorphism. Thus the theory of algebraically compact groups runs, in many respects, parallel to the theory of injective groups, a fact that was first pointed out by Maranda [1].]

Published: Jun 12, 2015

Keywords: Compact Group; Exchange Property; Continuous Homomorphism; Direct Decomposition; Complete Group

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