A remark on locally compact abelian groups. (English) Zbl 0063.03693

From the text: It has recently been shown by P. R. Halmos [Bull. Am. Math. Soc. 50, 877–878 (1944; Zbl 0061.04404)] that there exists a compact topological group which is algebraically isomorphic to the additive group of the real line, an example being given by the character group of the discrete additive group of the rationals. Exploiting his argument a bit further it is easy to see that the most general such example is the direct sum of \(\aleph\) replicas of the one already given where \(\aleph\) is a cardinal such that \(2^\aleph\le C\). This having been observed it naturally occurs to one to ask for the most general locally compact topological group with the algebraic structure in question. It is the purpose of the present note to give a complete answer to this question. (For details see the author’s theorem too lengthy to be stated here.)


22B05 General properties and structure of LCA groups


Zbl 0061.04404
Full Text: DOI


[1] Paul R. Halmos, Comment on the real line, Bull. Amer. Math. Soc. 50 (1944), 877 – 878. · Zbl 0061.04404
[2] L. Pontrjagin, Topological groups, Princeton, 1939. · JFM 65.0872.02
[3] A. Weil, L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, no. 869, Paris, 1940. · Zbl 0063.08195
[4] Leo Zippin, Countable torsion groups, Ann. of Math. (2) 36 (1935), no. 1, 86 – 99. · Zbl 0011.10401 · doi:10.2307/1968666
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