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A remark on free topological groups with no small subgroups

Published online by Cambridge University Press:  09 April 2009

H. B. Thompson
H. B. Thompson
Affiliation:
Flinders University, South Australia
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For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 4 , December 1974 , pp. 482 - 484
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Abels, Herbert, ‘Normen auf freien topologischen Gruppen’, Math. Z. 129 (1962), 25–42.Google Scholar
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[4]Kurosh, A. G., ‘Theory of Groups’, Volume 1 (Fund. Math., New York 1960).Google Scholar
[5]Morris, Sidney A., ‘Quotient groups of topological groups with no small subgroups’, Proc. Amer. Math. Soc. 31 (1972), 625–626.CrossRefGoogle Scholar
[6]Morris, Sidney A. and Thompson, H. B., ‘Free topological groups with no small subgroups’, (to appear).Google Scholar
[7]Morris, Sidney A. and Thompson, H. B., ‘Invariant metrics on free topological groups’, Bull. Austral. Math. Soc. 9 (1973), 83–88.CrossRefGoogle Scholar