Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T03:03:43.219Z Has data issue: false hasContentIssue false
  • English
  • Français

The construction of groups in models of set theory that fail the Axiom of Choice

Published online by Cambridge University Press:  17 April 2009

J.L. Hickman
J.L. Hickman
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to show that a well-known method for constructing “queer” sets in models of ZF set theory is also applicable to certain algebraic structures. An infinite set is called “quasi-minimal” if every subset of it is either finite or cofinite. In Section 1 I set out the two systems of set theory to be used in this paper, and illustrate the technique in its most fundamental form by constructing a model of set theory containing a quasi-minimal set. In Section 2 I show that by choosing the parameters appropriately, one can use this technique to obtain models of set theory containing groups whose carriers are quasi-minimal. In the third section various independence results are deduced from the existence of such models: in particular, it is shown that it is possible in ZF set theory to have an infinite group that satisfies both the ascending and descending chain conditions. The quasi-minimal groups constructed in Section 2 were all elementary abelian; in Section 4 it is shown that this was not just chance, but that in fact all quasi-minimal groups must be of this type. Finally in Section 5 permutations and permutation groups on quasi-minimal sets are examined.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 2 , April 1976 , pp. 199 - 232
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Drake, Frank R., Set theory: an introduction to large cardinals (Studies in Logic and the Foundations of Mathematics, 76. North-Holland, Amsterdam, London; American Elsevier, New York; 1974).Google Scholar
[2]Hickman, J.L. and Neumann, B.H., “A question of Babai on groups”, Bull. Austral. Math. Soc. 13 (1975), 355–368.CrossRefGoogle Scholar
[3]Jech, Thomas L., The Axiom of Choice (Studies in Logic and the Foundations of Mathematics, 75. North Holland, Amsterdam, London; American Elsevier, New York; 1973).Google Scholar
[4]Miller, G.A., “On an important theorem with respect to the operation groups of order p α, p being any prime number”, Messenger Math. 27 (1898), 119–121; see also, The collected works of George Abram Miller, Volume I, 303–304 (University of Illinois, Urbana, Illinois, 1935).Google Scholar
[5]Neumann, B.H., Private communication.Google Scholar
[6]Zassenhaus, Hans J., The theory of groups, second edition (Chelsea, New York, 1958).Google Scholar