Free topological groups with no small subgroups

Morris, Sidney A.; Thompson, H. B.

doi:10.1090/S0002-9939-1974-0352318-3

Free topological groups with no small subgroups
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by Sidney A. Morris and H. B. Thompson PDF

Proc. Amer. Math. Soc. 46 (1974), 431-437 Request permission

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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