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Stabilizers of Trivial Ideals

Stabilizers of Trivial Ideals JACINTA COVINGTON AND ALAN H. MEKLER 1. Introduction In papers by Semmes [6], Macpherson and Neumann [4] and Brazil, Covington, Penttila, Praeger and Woods [2], it has been shown that stabilizers of filters (equivalently, stabilizers of ideals) play a crucial role in the study of maximal subgroups of infinite symmetric groups. Semmes proved that if if is a maximal subgroup of S = Sym (Q), where Q is a set of infinite cardinality K and H contains the pointwise stabilizer of some set A with |A| < K, then H is the stabilizer of a filter. This was investigated further in [2], where it was shown that if H is a maximal subgroup containing the pointwise stabilizer of some set A with |A | = K, then H is the stabilizer of a quasiideal. This leads to the result that any such subgroup is either the almost stabilizer of a partition of Q into finitely many parts or the stabilizer of an ideal. The ideals obtained in these papers are all nontrivial ideals, that is, they contain some set of cardinality K. In [2] it was shown that if a maximal subgroup of S is the stabilizer of a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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References (6)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/25.4.337
Publisher site
See Article on Publisher Site

Abstract

JACINTA COVINGTON AND ALAN H. MEKLER 1. Introduction In papers by Semmes [6], Macpherson and Neumann [4] and Brazil, Covington, Penttila, Praeger and Woods [2], it has been shown that stabilizers of filters (equivalently, stabilizers of ideals) play a crucial role in the study of maximal subgroups of infinite symmetric groups. Semmes proved that if if is a maximal subgroup of S = Sym (Q), where Q is a set of infinite cardinality K and H contains the pointwise stabilizer of some set A with |A| < K, then H is the stabilizer of a filter. This was investigated further in [2], where it was shown that if H is a maximal subgroup containing the pointwise stabilizer of some set A with |A | = K, then H is the stabilizer of a quasiideal. This leads to the result that any such subgroup is either the almost stabilizer of a partition of Q into finitely many parts or the stabilizer of an ideal. The ideals obtained in these papers are all nontrivial ideals, that is, they contain some set of cardinality K. In [2] it was shown that if a maximal subgroup of S is the stabilizer of a

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jul 1, 1993

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