Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-22T14:41:49.476Z Has data issue: false hasContentIssue false
  • English
  • Français

Locally compact topologies on abelian groups

Published online by Cambridge University Press:  24 October 2008

Sidney A. Morris
Sidney A. Morris
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, Vic. 3083, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that an abelian group admits a non-discrete locally compact group topology if and only if it has a subgroup algebraically isomorphic to the group of p-adic integers or to an infinite product of non-trivial finite cyclic groups. It is also proved that an abelian group admits a non-totally-disconnected locally compact group topology if and only if it has a subgroup algebraically isomorphic to the group of real numbers. Further, if an abelian group admits one non-totally-disconnected locally compact group topology then it admits a continuum of such topologies, no two of which yield topologically isomorphic topological groups.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 2 , March 1987 , pp. 233 - 235
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Armacost, D. L.. The Structure of Locally Compact Abelian Groups (Marcel Dekker, 1981).Google Scholar
[2]Corwin, L.. Uniqueness of topology for the p-adic integers. Proc. Amer. Math. Soc. 55 (1976), 432–434.Google Scholar
[3]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, vol. I (Springer-Verlag, 1963).Google Scholar
[4]Morris, S. A.. Pontryagin Duality and the Structure of Locally Compact Abelian Groups (Cambridge University Press, 1977).CrossRefGoogle Scholar
[5]Soundararajan, T.. The topological group of the p-adic integers. Publ. Math. Debrecen 16 (1969), 75–78.CrossRefGoogle Scholar