Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22A05, 20E05

Retrieve articles in all journals with MSC: 22A05, 20E05


Additional Information

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