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A Schreier theorem for free topological groups

Published online by Cambridge University Press:  17 April 2009

Peter Nickolas
Peter Nickolas
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
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Abstract

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M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 1 , August 1975 , pp. 121 - 127
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Brown, Ronald, “Some non-projective subgroups of free topological groups”, Proc. Amer. Math. Soc. (to appear).Google Scholar
[2]Brown, R. and Hardy, J.P.L., “Subgroups of free topological groups and free topological products of topological groups”, J. London Math. Soc. (to appear).Google Scholar
[3]Graev, M.I., “Free topological groups”, Amer. Math. Soc. Transl. 35 (1951), 61 pp.; reprinted Amer. Math. Soc. Transl. (l) 8 (1962), 305–364.Google Scholar
[4]Hall, Marshall Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
[5]Hardy, J.P.L., “Topological groupoids: coverings and universal constructions”, (PhD thesis, University College of North Wales, Bangor, Wales, 1974).Google Scholar
[6]Higgins, Philip J., Notes on categories and groupoids (van Nostrand Reinhold, London, New York, Cincinnati, Toronto, Melbourne, 1971).Google Scholar
[7]Mack, J., Morris, Sidney A. and Ordman, E.T., “Free topological groups and the projective dimension of a locally compact abelian group”, Proc. Amer. Math. Soc. 40 (1973), 303–308.Google Scholar
[8]Madison, Bernard L., “Congruences in topological semigroups”, Second Florida Symposium on automata and semigroups (University of Florida, Tampa, 1971).Google Scholar
[9]Markov, A.A., “On free topological groups”, Amer. Math. Soc. Transl. 30 (1950), 11–88; reprinted Amer. Math. Soc. Transl. (1) 8 (1962), 195–272.Google Scholar