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Supplements of bounded permutation groups

Published online by Cambridge University Press:  12 March 2014

Stephen Bigelow
Stephen Bigelow*
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA, E-mail: [email protected]
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Abstract

Let λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group Bλ(Ω), or simply Bλ, is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of Bλ if BλG is the full symmetric group on Ω.

In [7], Macpherson and Neumann claimed to have classified all supplements of bounded permutation groups. Specifically, they claimed to have proved that a group G acting on the set Ω is a supplement of Bλ if and only if there exists Δ ⊂ Ω with ∣Δ∣ < λ such that the setwise stabiliser G{Δ} acts as the full symmetric group on Ω ∖ Δ. However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developed pcf theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 89 - 102
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1]Burke, M. R. and Magidor, M., Shelah's pcf theory and its applications, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 207–254.CrossRefGoogle Scholar
[2]de Bruijin, N. G., A theorem on choice functions, Koninklijke Nederlandse Akademie van Wetenschappen Indagationes Mathematicae, vol. 19 (1957), pp. 409–411.Google Scholar
[3]Dodd, A. and Jensen, R. B., The core model, Annals of Mathematical Logic, vol. 20 (1981), pp. 43–75.CrossRefGoogle Scholar
[4]Dodd, A. and Jensen, R. B., The covering lemma for K, Annals of Mathematical Logic, vol. 22 (1982), pp. 1–30.CrossRefGoogle Scholar
[5]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[6]Kunen, K., Set theory: An introduction to independence proofs, North Holland, Amsterdam, 1980.Google Scholar
[7]Macpherson, H. D. and Neumann, Peter M., Subgroups of infinite symmetric groups, Journal of the London Mathematical Society, vol. 42 (1990), no. 2, pp. 64–84.CrossRefGoogle Scholar
[8]Magidor, M., On the singular cardinals problem I, Israel Journal of Mathematics, vol. 28 (1977), pp. 1–31.CrossRefGoogle Scholar
[9]Scott, W. R., Group theory, Prentice-Hall, New Jersey, 1964.Google Scholar
[10]Semmes, Stephen W., Infinite symmetric groups, maximal subgroups, and filters, Abstracts of the American Mathematical Society, vol. 69 (1982), p. 38, preliminary report.Google Scholar
[11]Shelah, S., The singular cardinals problem: Independence results, Proceedings of a symposium on set theory, Cambridge 1978 (Mathias, A., editor), London Mathematical Society Lecture Notes Series, no. 87, Cambridge University Press, Cambridge and New York, 1983, pp. 116–134.Google Scholar
[12]Shelah, S., Cardinal arithmetic for skeptics, Bulletin of the American Mathematical Society, vol. 26 (1992), pp. 197–210.CrossRefGoogle Scholar
[13]Shelah, S., Cardinal arithmetic, Oxford University Press, New York, 1994.CrossRefGoogle Scholar