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Free topological groups with no small subgroups


Authors: Sidney A. Morris and H. B. Thompson
Journal: Proc. Amer. Math. Soc. 46 (1974), 431-437
MSC: Primary 22A05; Secondary 20E05
DOI: https://doi.org/10.1090/S0002-9939-1974-0352318-3
MathSciNet review: 0352318
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Abstract: The first author has shown that a quotient group of a topological group with no small subgroups can have small subgroups, thus answering a question of Kaplansky in the negative. The argument relied on showing that a free abelian topological group on any metric space has no small subgroups. Consequently any abelian metric group is a quotient of a group with no small subgroups. However metric groups with small subgroups exist in profusion! It is shown here that a necessary and sufficient condition for a free (free abelian) topological group on a topological space $ X$ to have no small subgroups is that $ X$ admits a continuous metric. Hence any topological group which admits a continuous metric is a quotient group of a group with no small subgroups.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0352318-3
Article copyright: © Copyright 1974 American Mathematical Society

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