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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Abelian subgroups of topological groups


Authors: Siegfried K. Grosser and Wolfgang N. Herfort
Journal: Trans. Amer. Math. Soc. 283 (1984), 211-223
MSC: Primary 22A05; Secondary 22D05
MathSciNet review: 735417
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Abstract: In [1] Šmidt's conjecture on the existence of an infinite abelian subgroup in any infinite group is settled by counterexample. The well-known Hall-Kulatilaka Theorem asserts the existence of an infinite abelian subgroup in any infinite locally finite group. This paper discusses a topological analogue of the problem. The simultaneous consideration of a stronger condition--that centralizers of nontrivial elements be compact--turns out to be useful and, in essence, inevitable. Thus two compactness conditions that give rise to a profinite arithmetization of topological groups are added to the classical list (see, e.g., [13 or 4]).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0735417-4
PII: S 0002-9947(1984)0735417-4
Keywords: Compactness conditions, profinite theory, Lie groups, Moore groups
Article copyright: © Copyright 1984 American Mathematical Society