Premium
Stabilizers of Trivial Ideals
Author(s) -
Covington Jacinta,
Mekler Alan H.
Publication year - 1993
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/25.4.337
Subject(s) - mathematics , citation , library science , combinatorics , algebra over a field , computer science , pure mathematics
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 nontrivial ideal, then this is the unique such nontrivial ideal.